---
title: "Khosla: intermediate pile pressure (L1 30 m)"
subject: "Irrigation and Drainage Engineering"
chapter: 5
key: "semester_6/irrigation_and_drainage_engineering:khosla-headworks-levels-table-intermediate-pile"
source: https://www.bctengineer.com/q/semester_6/irrigation_and_drainage_engineering/khosla-headworks-levels-table-intermediate-pile
site: BCT Engineering
---

# Khosla: intermediate pile pressure (L1 30 m)

Irrigation and Drainage Engineering · Chapter 5: Diversion Headworks

- Asked in: 2068 Chaitra (old course), 7+3 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-intermediate-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, compute the pressure at the intermediate pile section: floor and pile interface and bottom end of pile. Check the exit gradient of the hydraulic structure. The safe exit gradient of the soil is 1/6.

[Figure: weir with 4:1 upstream slope; Level A upstream water level, B upstream floor, C crest, D downstream floor, E bottom of upstream pile, F upstream floor bottom, G bottom of intermediate pile, H, I downstream floor levels, J bottom of downstream pile; L1 and L2 horizontal distances]

| 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 30 m and 60 m respectively. Correction factor for slope is 3.3. Use analytical method. Make suitable assumption if necessary.

## Short answer

Intermediate pile pressures: E2=12.21m (64.26%), D2=11.54m (60.76%), C2=11.26m (59.26%). Exit gradient = 0.356 > 1/6 (0.167) — unsafe; deepen downstream pile or lengthen floor.

## 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 $L_1 = 30$ m, d/s pile at $L_1 + L_2 = 90$ m. Slope correction factor 3.3% (4:1 slope), multiplied by (slope length / $b'$) with $b' = 30$ m (both slopes lie between the u/s and intermediate piles): u/s slope 12 m (up slope, $-$ at $C_1$) and d/s slope 16 m (down slope, $+$ at $E_2$). Floor thickness at the intermediate pile is not given and is neglected there.

#### 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), $b = 90$ 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 | 18.000 | 9.514 | 21.02 | 14.73 |
| D/s pile | 6.00 | 15.000 | 8.017 | 22.98 | 16.07 |

| Intermediate pile | d (m) | α1 | α2 | λ1 | λ | φE (%) | φD (%) | φC (%) |
|---|---|---|---|---|---|---|---|---|
| at 30 m | 3.00 | 10.000 | 20.000 | -4.988 | 15.037 | 63.04 | 60.76 | 58.54 |

#### 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 | 85.27 | - | 85.27 | 16.20 |
| C1 | 78.98 | thk +1.26, int +0.69, slope -1.32 | 79.61 | 15.13 |
| E2 | 63.04 | int -0.54, slope +1.76 | 64.26 | 12.21 |
| D2 | 60.76 | - | 60.76 | 11.54 |
| C2 | 58.54 | int +0.72 | 59.26 | 11.26 |
| E3 | 22.98 | thk -1.15, int -0.31 | 21.52 | 4.09 |
| D3 | 16.07 | - | 16.07 | 3.05 |
| 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, e.g. at $C_2$: $D = 142-135 = 7$ m, $d = 3$ m, $b' = 60$ m); slope = slope correction. The D points are not corrected.

#### Exit gradient

$$
G_E = \frac{H}{d}\cdot\frac{1}{\pi\sqrt{\lambda}} = \frac{19.00}{6.00}\times\frac{1}{\pi\sqrt{8.017}} = 0.3560\ (\approx 1/2.8)
$$

Permissible $= 1/6 = 0.1667$. Not safe; deepen the d/s pile or lengthen the floor.

#### Hydraulic gradient line

Bars show the corrected pressure (each # is 2.5%):
```text
 E1  100.0% |########################################
 D1   85.3% |##################################
 C1   79.6% |################################
 E2   64.3% |##########################
 D2   60.8% |########################
 C2   59.3% |########################
 E3   21.5% |#########
 D3   16.1% |######
 C3    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 of one pile to E of the next).

**Answer:** At the intermediate pile: floor-pile interfaces $E_2 = 64.3\%$ and $C_2 = 59.3\%$ of $H$ (12.21 m and 11.26 m); bottom end of the pile $D_2 = 60.8\%$ (11.54 m). Exit gradient $G_E = 0.356$, more than 1/6 = 0.167, 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. With no flow, the downstream water level is taken at the d/s floor level (141 m) and the u/s water level is 160 m. What head H is used for the percentage pressures?
   A. 14 m
   B. 17 m
   C. 19 m
   D. 22 m
   Answer: C. H = 160 - 141 = 19 m. The percentage pressures are fractions of this head. (Key point: Head with no flow)

2. The intermediate pile bottom G is at 139 m and the floor level there is 142 m. What depth d is used in Khosla's constants for this pile?
   A. 4 m
   B. 1 m
   C. 5 m
   D. 3 m
   Answer: D. d is measured from the floor level to the pile bottom: 142 - 139 = 3 m. (Key point: Pile depth from levels)

3. How does the hydraulic gradient line behave across a pile and along the floor?
   A. Falls linearly along the pile and is flat along the floor
   B. Drops vertically across each pile and is straight along the floor between pile junctions
   C. Drops only at the downstream pile
   D. Is a single straight line from E1 to C3
   Answer: B. The HGL is drawn as vertical drops at the piles (E to D to C) joined by straight lines on the floor. (Key point: Shape of the HGL)

4. The exit gradient G_E = (H/d)(1/(pi sqrt(lambda))) computed at the d/s pile is 0.356 against a safe value of 1/6. What is concluded?
   A. Cannot be judged without floor thickness
   B. Safe because the d/s pile is the deepest
   C. Not safe; deepen the d/s pile or lengthen the floor
   D. Safe since 0.356 is greater than 0.167
   Answer: C. G_E must be less than the permissible 0.167. 0.356 exceeds it, so piping is a risk. (Key point: Exit gradient check)

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