---
title: "Khosla: thickness and exit gradient, 105 m head"
subject: "Irrigation and Drainage Engineering"
chapter: 5
key: "semester_6/irrigation_and_drainage_engineering:khosla-thickness-exit-gradient-fig-105"
source: https://www.bctengineer.com/q/semester_6/irrigation_and_drainage_engineering/khosla-thickness-exit-gradient-fig-105
site: BCT Engineering
---

# Khosla: thickness and exit gradient, 105 m head

Irrigation and Drainage Engineering · Chapter 5: Diversion Headworks

- Asked in: 2062 Baisakh (old course), 16 marks
- Video: [Problem on Khosla Theory.](https://www.youtube.com/watch?v=2NEHgwXCa-U) — Civil Giri
- 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-thickness-exit-gradient-fig-105
- Whole chapter: https://www.bctengineer.com/md/semester_6/irrigation_and_drainage_engineering/chapter-5

## Question

Check the thickness and exit gradient from Khosla theory. The safe exit gradient is 0.2.

[Figure: water level 105 m; levels 101 m, 100 m, 100 m (downstream), 98 m (upstream pile bottom), 97.5 m (intermediate pile bottom), 97 m (downstream pile bottom); 2 m floor depression; horizontal distances 5 m, 5 m and 20 m]

## Short answer

Corrected pressure heads: C1=3.77 m, E2=3.88 m, C2=3.34 m, E3=1.34 m. Required floor thickness: u/s pile 1.98 m, intermediate pile u/s face 2.77 m and d/s face 2.39 m, d/s pile 0.96 m. Exit gradient = 0.226, unsafe (>0.2).

## Answer

#### Data and assumptions

Reading of the figure: water level 105 m; with no flow the d/s water level is the d/s floor level 100 m, so $H = 5$ m. U/s pile at 0 m: floor 101 m, bottom 98 m ($d = 3$ m). Intermediate pile at 5 m: floor 100 m, bottom 97.5 m ($d = 2.5$ m). D/s pile at 30 m ($5 + 5 + 20$): floor 100 m, bottom 97 m ($d = 3$ m). Floor thickness is not known, so the required thickness is computed and the thickness correction is neglected. $G = 2.4$. Safe exit gradient 0.2.

#### 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), $b = 30$ 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 | 3.00 | 10.000 | 5.525 | 27.98 | 19.45 |
| D/s pile | 3.00 | 10.000 | 5.525 | 27.98 | 19.45 |

| Intermediate pile | d (m) | α1 | α2 | λ1 | λ | φE (%) | φD (%) | φC (%) |
|---|---|---|---|---|---|---|---|---|
| at 5 m | 2.50 | 2.000 | 10.000 | -3.907 | 6.143 | 79.45 | 71.94 | 65.69 |

#### Pressures at the key points

Pressure is expressed as a percentage of the head $H = 5.00$ m (no flow).

| Point | Simple φ (%) | Corrections (%) | Corrected φ (%) | Pressure head (m) |
|---|---|---|---|---|
| E1 (top of u/s pile) | 100.00 | 0 | 100.00 | 5.00 |
| D1 | 80.55 | - | 80.55 | 4.03 |
| C1 | 72.02 | int +3.44 | 75.47 | 3.77 |
| E2 | 79.45 | int -1.80 | 77.65 | 3.88 |
| D2 | 71.94 | - | 71.94 | 3.60 |
| C2 | 65.69 | int +1.21 | 66.90 | 3.34 |
| E3 | 27.98 | int -1.10 | 26.87 | 1.34 |
| D3 | 19.45 | - | 19.45 | 0.97 |
| C3 (d/s end) | 0 | 0 | 0 | 0 |

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). The D points are not corrected. ($E_2$ comes out slightly above $C_1$ because the two piles are only 5 m apart.)

#### 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 straight between the corrected key points along the floor).

| Location | x (m) | φ (%) | HG level (m) | Floor top (m) | Unbalanced head (m) | Required t (m) | Provided t (m) | Check |
|---|---|---|---|---|---|---|---|---|
| C1 (u/s pile) | 0 | 75.47 | 103.77 | 101.00 | 2.77 | 1.98 | - | provide at least 1.98 |
| E2 (int. pile, u/s face) | 5 | 77.65 | 103.88 | 100.00 | 3.88 | 2.77 | - | provide at least 2.77 |
| C2 (int. pile, d/s face) | 5 | 66.90 | 103.34 | 100.00 | 3.34 | 2.39 | - | provide at least 2.39 |
| E3 (d/s pile) | 30 | 26.87 | 101.34 | 100.00 | 1.34 | 0.96 | - | provide at least 0.96 |

#### Exit gradient

$$
G_E = \frac{H}{d}\cdot\frac{1}{\pi\sqrt{\lambda}} = \frac{5.00}{3.00}\times\frac{1}{\pi\sqrt{5.525}} = 0.2257\ (\approx 1/4.4)
$$

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

## 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 100 m. What is H?
   A. 3 m
   B. 4 m
   C. 5 m
   D. 8 m
   Answer: C. H = 105 - 100 = 5 m. (Key point: Head with no flow)

2. Why is the thickness correction neglected?
   A. It is negligible for all floors
   B. The piles are too shallow
   C. Floor thickness is not known; required thickness is computed instead
   D. Slope replaces it
   Answer: C. With unknown provided thickness, only the required thickness is found. (Key point: Thickness correction omitted)

3. Required thickness is 1.86 m, 2.32 m and 1.00 m at three locations. The floor should be at least
   A. 1.00 m
   B. 1.86 m everywhere
   C. 2.32 m where uplift is greatest
   D. The average of the three
   Answer: C. The thickness needed is highest at the intermediate pile, 2.32 m, because the uplift head is highest there. (Key point: Governing thickness)

4. G_E = 0.2257 against 0.2. What is concluded?
   A. Safe
   B. Safe if G is 2.4
   C. Not safe
   D. Safe since lambda is large
   Answer: C. 0.2257 exceeds 0.2, so piping is a risk. (Key point: Exit gradient verdict)

## Related in Irrigation and Drainage Engineering

- [Khosla: uplift, exit gradient 0.15, floor at A-D](https://www.bctengineer.com/question.md?key=semester_6%2Firrigation_and_drainage_engineering%3Akhosla-uplift-exit-gradient-015-abcd) — Irrigation and Drainage Engineering, chapter 5
- [Khosla: HGL and exit gradient, pond 102.50](https://www.bctengineer.com/question.md?key=semester_6%2Firrigation_and_drainage_engineering%3Akhosla-hgl-weir-profile-pond1025) — Irrigation and Drainage Engineering, chapter 5
- [Khosla: uplift at A, B, thickness, exit gradient 0.2](https://www.bctengineer.com/question.md?key=semester_6%2Firrigation_and_drainage_engineering%3Akhosla-uplift-floor-thickness-fig-103) — 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: 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: 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

## Related in other subjects

- [Water pipe thickness from pressure head](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Awater-pipe-thickness-pressure-head) — Strength of Materials, chapter 5