--- title: "Khosla: end pile pressure (L1 20 m)" subject: "Irrigation and Drainage Engineering" chapter: 5 key: "semester_6/irrigation_and_drainage_engineering:khosla-headworks-levels-table-end-pile" source: https://www.bctengineer.com/q/semester_6/irrigation_and_drainage_engineering/khosla-headworks-levels-table-end-pile site: BCT Engineering --- # Khosla: end pile pressure (L1 20 m) Irrigation and Drainage Engineering · Chapter 5: Diversion Headworks - Asked in: 2068 Baisakh (old course), 6+2+2 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-headworks-levels-table-end-pile - Whole chapter: https://www.bctengineer.com/md/semester_6/irrigation_and_drainage_engineering/chapter-5 ## Question For the following irrigation headworks and respective data, i) Compute the pressure at the end pile section: floor and pile interface, bottom end of pile. ii) Check whether the floor thickness provided at the end is sufficient or not? iii) Check the exit gradient of the hydraulic structure. The safe exit gradient of the soil is 1/6. [Figure: same weir profile as the table below with 4:1 upstream slope; levels A to J, distances L1 and L2] | Reference Point | Level, masl | Reference Point | Level, masl | |---|---|---|---| | A | 160 | F | 142 | | B | 143 | G | 139 | | C | 146 | H | 140 | | D | 142 | I | 141 | | E | 138 | J | 135 | L1 and L2 are 20 m and 60 m respectively. Use analytical method or Khosla's curve. Make suitable assumptions if necessary. ## Short answer End pile (downstream) pressures: E3=4.33m (22.77%), D3=3.23m (16.99%), C3=0. Floor at downstream end (1.0m provided) requires 3.09m — unsafe. Exit gradient = 0.376 > 1/6 (0.167) — unsafe; deepen downstream pile. ## Answer #### Data and assumptions Reading of the table: A = u/s water level 160 m; B = u/s floor level 143 m; C = crest 146 m; D = d/s floor level (at the intermediate pile) 142 m; E = bottom of u/s pile 138 m ($d_1 = 143-138 = 5$ m); F = bottom of the u/s floor 142 m (thickness 1.0 m); G = bottom of intermediate pile 139 m ($d_2 = 142-139 = 3$ m); I = d/s floor level 141 m and H = bottom of the d/s floor 140 m (thickness 1.0 m); J = bottom of d/s pile 135 m ($d_3 = 141-135 = 6$ m). With no flow the d/s water level is the d/s floor level 141 m, so $H = 160 - 141 = 19$ m. U/s pile at 0 m, intermediate pile at $L_1 = 20$ m, d/s pile at $L_1 + L_2 = 80$ m. The slope correction is not given and is neglected (the slopes are far from the end pile, so it would not change $E_3$ or $D_3$). #### 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), $b = 80$ 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): $\lambda_1 = \dfrac{\sqrt{1+\alpha_1^2}-\sqrt{1+\alpha_2^2}}{2}$, $\lambda = \dfrac{\sqrt{1+\alpha_1^2}+\sqrt{1+\alpha_2^2}}{2}$, $\phi_E = \dfrac{1}{\pi}\cos^{-1}\dfrac{\lambda_1-1}{\lambda}$, $\phi_D = \dfrac{1}{\pi}\cos^{-1}\dfrac{\lambda_1}{\lambda}$, $\phi_C = \dfrac{1}{\pi}\cos^{-1}\dfrac{\lambda_1+1}{\lambda}$. | Pile | d (m) | α | λ | φE (%) | φD (%) | |---|---|---|---|---|---| | U/s pile | 5.00 | 16.000 | 8.516 | 22.27 | 15.58 | | D/s pile | 6.00 | 13.333 | 7.185 | 24.34 | 16.99 | | Intermediate pile | d (m) | α1 | α2 | λ1 | λ | φE (%) | φD (%) | φC (%) | |---|---|---|---|---|---|---|---|---| | at 20 m | 3.00 | 6.667 | 20.000 | -6.642 | 13.383 | 69.34 | 66.53 | 63.85 | #### Pressures at the key points Pressure is expressed as a percentage of the head $H = 19.00$ m (no flow). | Point | Simple φ (%) | Corrections (%) | Corrected φ (%) | Pressure head (m) | |---|---|---|---|---| | E1 (top of u/s pile) | 100.00 | 0 | 100.00 | 19.00 | | D1 | 84.42 | - | 84.42 | 16.04 | | C1 | 77.73 | thk +1.34, int +0.96 | 80.03 | 15.20 | | E2 | 69.34 | int -0.74 | 68.60 | 13.03 | | D2 | 66.53 | - | 66.53 | 12.64 | | C2 | 63.85 | int +0.81 | 64.66 | 12.29 | | E3 | 24.34 | thk -1.22, int -0.35 | 22.77 | 4.33 | | D3 | 16.99 | - | 16.99 | 3.23 | | C3 (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, $19\sqrt{D/b'}\,\dfrac{d+D}{b}$, 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; at $E_3$: $D = 141-139 = 2$ m, $d = 6$ m, $b' = 60$ 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 $G = 2.40$. | Location | x (m) | φ (%) | HG level (m) | Floor top (m) | Unbalanced head (m) | Required t (m) | Provided t (m) | Check | |---|---|---|---|---|---|---|---|---| | E3 (d/s end) | 80 | 22.77 | 145.33 | 141.00 | 4.33 | 3.09 | 1.00 | not safe | #### Exit gradient $$ G_E = \frac{H}{d}\cdot\frac{1}{\pi\sqrt{\lambda}} = \frac{19.00}{6.00}\times\frac{1}{\pi\sqrt{7.185}} = 0.3760\ (\approx 1/2.7) $$ Permissible $= 1/6 = 0.1667$. Not safe; deepen the d/s pile or lengthen the floor. **Answer:** At the d/s (end) pile: floor-pile interface $E_3 = 22.8\%$ of $H$ (4.33 m); bottom end of the pile $D_3 = 17.0\%$ (3.23 m). Required floor thickness at the end $= 3.09$ m against 1.0 m provided, so the provided thickness is **not sufficient**. Exit gradient $G_E = 0.376$ against 1/6 = 0.167, so the exit gradient is **not safe** either (a deeper d/s pile or a longer floor is needed). ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. How is the required floor thickness at a point found from the HGL? A. t = h x (G-1) B. t = h/(G-1), where h is the unbalanced head (HG level minus floor top level) C. t = h/(G+1) D. t = H/G for the whole structure Answer: B. Uplift head h is balanced by the submerged weight of the floor, so t = h/(G-1). (Key point: Required floor thickness formula) 2. At the d/s end the required thickness is 3.14 m while 1.0 m is provided. What is the conclusion? A. Sufficient only if G is increased to 3.14 B. The check is not needed at the end C. Provided thickness is not sufficient D. Provided thickness is sufficient Answer: C. Provided 1.0 m is less than the required 3.14 m, so the floor is unsafe against uplift. (Key point: Thickness check result) 3. The exit gradient is calculated using which pile? A. The u/s pile, with d = 5 m B. The average of all three piles C. The intermediate pile with d = 3 m D. The d/s pile, with its depth d = 6 m and lambda Answer: D. The exit gradient is critical at the d/s end where water leaves, so d and lambda of the d/s pile are used. (Key point: Exit gradient location) 4. G_E = 0.376 against permissible 1/6 (0.167). Which remedy fits? A. Do nothing since the floor is safe B. Increase d/s pile depth or lengthen the floor C. Reduce the floor length D. Reduce the d/s pile depth Answer: B. G_E falls with greater d or longer floor (larger lambda), so either remedy brings it below the limit. (Key point: Exit gradient verdict) ## Related in Irrigation and Drainage Engineering - [Khosla: intermediate pile pressure (L1 30 m)](https://www.bctengineer.com/question.md?key=semester_6%2Firrigation_and_drainage_engineering%3Akhosla-headworks-levels-table-intermediate-pile) — Irrigation and Drainage Engineering, chapter 5 - [Khosla: thickness and exit gradient, 105 m head](https://www.bctengineer.com/question.md?key=semester_6%2Firrigation_and_drainage_engineering%3Akhosla-thickness-exit-gradient-fig-105) — Irrigation and Drainage Engineering, chapter 5 - [Khosla: uplift, HGL, thickness, exit gradient 1/5](https://www.bctengineer.com/question.md?key=semester_6%2Firrigation_and_drainage_engineering%3Akhosla-hgl-thickness-gradient-1-in) — Irrigation and Drainage Engineering, chapter 5 - [Khosla: barrage HGL and floor thickness](https://www.bctengineer.com/question.md?key=semester_6%2Firrigation_and_drainage_engineering%3Akhosla-barrage-hgl-thickness-pond1135) — Irrigation and Drainage Engineering, chapter 5 - [Khosla: two piles, floor at A, B, C, exit 0.2](https://www.bctengineer.com/question.md?key=semester_6%2Firrigation_and_drainage_engineering%3Akhosla-two-piles-thickness-abc-02) — Irrigation and Drainage Engineering, chapter 5 - [Khosla: cistern floor thickness from corrected phi](https://www.bctengineer.com/question.md?key=semester_6%2Firrigation_and_drainage_engineering%3Akhosla-corrected-phi-barrage-cistern) — Irrigation and Drainage Engineering, chapter 5