---
title: "S-curve 3-h UH and flood from two storms"
subject: "Engineering Hydrology"
chapter: 5
key: "semester_5/engineering_hydrology:scurve-2h-3h-flood-8cm-7cm"
source: https://www.bctengineer.com/q/semester_5/engineering_hydrology/scurve-2h-3h-flood-8cm-7cm
site: BCT Engineering
---

# S-curve 3-h UH and flood from two storms

Engineering Hydrology · Chapter 5: Hydrograph Analysis

- Asked in: 2075 Asoj, 10 marks
- Video: [Numerical on 'S' Curve Hydrograph (SSCJE/GATE/ESE/PSU's/DFCCIL/Railways)-Civil Engineering](https://www.youtube.com/watch?v=ChzzFVxNlm0) — Engineering Ace- By Varun Sir
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/engineering_hydrology/scurve-2h-3h-flood-8cm-7cm
- Whole chapter: https://www.bctengineer.com/md/semester_5/engineering_hydrology/chapter-5

## Question

The ordinates of a 2-h UH are given below. Derive the ordinates of a 3-h UH by S-curve method.

| Time (hr) | 0 | 2 | 4 | 6 | 8 | 10 | 12 | 14 | 16 | 18 | 20 | 22 | 24 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Ordinates of 2-h UH (m$^3$/s) | 0 | 25 | 100 | 160 | 190 | 170 | 110 | 70 | 30 | 20 | 15 | 6 | 0 |

Calculate the flood discharge of a storm of 3 h and 2 h rainfall of 8 cm and 7 cm respectively. Consider $\phi$-index 0.3 cm/hr and baseflow 10 m$^3$/s.

## Short answer

Derive3-h UH via S-curve: peak180 m³/s at8 h. Effective rain after φ=0.3 cm/h:7.1 cm over3 h, then6.4 cm over2 h. Flood hydrograph with baseflow10 m³/s peaks at2384 m³/s at10 h.

## Answer

#### Part 1: 3-h UH from the 2-h UH by the S-curve method
$S(t)=U_2(t)+S(t-2)$, and $U_3(t)=\dfrac{2}{3}\left[S(t)-S(t-3)\right]$ with $S(t-3)$ interpolated linearly.

| t (h) | $U_2$ | $S(t)$ | $S(t-3)$ | $U_3$ (m$^3$/s) |
|---|---|---|---|---|
| 0 | 0 | 0 | 0 | 0.00 |
| 2 | 25 | 25 | 0 | 16.67 |
| 4 | 100 | 125 | 12.5 | 75.00 |
| 6 | 160 | 285 | 75 | 140.00 |
| 8 | 190 | 475 | 205 | 180.00 |
| 10 | 170 | 645 | 380 | 176.67 |
| 12 | 110 | 755 | 560 | 130.00 |
| 14 | 70 | 825 | 700 | 83.33 |
| 16 | 30 | 855 | 790 | 43.33 |
| 18 | 20 | 875 | 840 | 23.33 |
| 20 | 15 | 890 | 865 | 16.67 |
| 22 | 6 | 896 | 882.5 | 9.00 |
| 24 | 0 | 896 | 893 | 2.00 |

The S-curve reaches equilibrium (896 m$^3$/s) at 22 h.

#### Part 2: Flood hydrograph
**Effective rainfall** ($\phi$ = 0.3 cm/h, assumed consecutive storms: 3 h of 8 cm, then 2 h of 7 cm):
- 3-h storm: $8 - 0.3\times3 = 7.1$ cm
- 2-h storm: $7 - 0.3\times2 = 6.4$ cm, starting at t = 3 h.

$$
Q(t)=10+7.1\,U_3(t)+6.4\,U_2(t-3)
$$
$U_3$ and $U_2$ are read at hourly intervals (linear interpolation of the S-curve; $U_2(t)=S(t)-S(t-2)$).

| t (h) | $U_3(t)$ | $U_2(t-3)$ | 7.1 $U_3$ | 6.4 $U_2$ | Q (m$^3$/s) |
|---|---|---|---|---|---|
| 0 | 0.0 | 0.0 | 0.0 | 0.0 | 10.0 |
| 1 | 8.3 | 0.0 | 59.2 | 0.0 | 69.2 |
| 2 | 16.7 | 0.0 | 118.3 | 0.0 | 128.3 |
| 3 | 50.0 | 0.0 | 355.0 | 0.0 | 365.0 |
| 4 | 75.0 | 12.5 | 532.5 | 80.0 | 622.5 |
| 5 | 120.0 | 25.0 | 852.0 | 160.0 | 1022.0 |
| 6 | 140.0 | 62.5 | 994.0 | 400.0 | 1404.0 |
| 7 | 170.0 | 100.0 | 1207.0 | 640.0 | 1857.0 |
| 8 | 180.0 | 130.0 | 1278.0 | 832.0 | 2120.0 |
| 9 | 183.3 | 160.0 | 1301.7 | 1024.0 | 2335.7 |
| 10 | 176.7 | 175.0 | 1254.3 | 1120.0 | 2384.3 |
| 11 | 150.0 | 190.0 | 1065.0 | 1216.0 | 2291.0 |
| 12 | 130.0 | 180.0 | 923.0 | 1152.0 | 2085.0 |
| 13 | 96.7 | 170.0 | 686.3 | 1088.0 | 1784.3 |
| 14 | 83.3 | 140.0 | 591.7 | 896.0 | 1497.7 |
| 15 | 56.7 | 110.0 | 402.3 | 704.0 | 1116.3 |
| 16 | 43.3 | 90.0 | 307.7 | 576.0 | 893.7 |
| 17 | 26.7 | 70.0 | 189.3 | 448.0 | 647.3 |
| 18 | 23.3 | 50.0 | 165.7 | 320.0 | 495.7 |
| 19 | 18.3 | 30.0 | 130.2 | 192.0 | 332.2 |
| 20 | 16.7 | 25.0 | 118.3 | 160.0 | 288.3 |
| 21 | 12.0 | 20.0 | 85.2 | 128.0 | 223.2 |
| 22 | 9.0 | 17.5 | 63.9 | 112.0 | 185.9 |
| 23 | 4.0 | 15.0 | 28.4 | 96.0 | 134.4 |
| 24 | 2.0 | 10.5 | 14.2 | 67.2 | 91.4 |
| 25 | 0.0 | 6.0 | 0.0 | 38.4 | 48.4 |
| 26 | 0.0 | 3.0 | 0.0 | 19.2 | 29.2 |
| 27 | 0.0 | 0.0 | 0.0 | 0.0 | 10.0 |

**Answer: The 3-h UH ordinates are 0, 16.7, 75, 140, 180, 176.7, 130, 83.3, 43.3, 23.3, 16.7, 9, 2 m$^3$/s at 0, 2, ..., 24 h. The flood hydrograph peaks at about 2384 m$^3$/s at 10 h and returns to the base flow of 10 m$^3$/s by about 27 h.**

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. The 3-h UH from the 2-h UH is:
   A. (3/2)[S(t)-S(t-3)]
   B. (2/3)[S(t)-S(t-3)]
   C. (2/3)[S(t)-S(t-2)]
   D. S(t)-S(t-3)
   Answer: B. The S-curve of 2-h UH is lagged 3 h and scaled by 2/3. (Key point: 3-h UH)

2. For the 3-h, 8 cm storm with phi = 0.3 cm/h, effective rainfall is:
   A. 7.1 cm
   B. 7.7 cm
   C. 8.0 cm
   D. 6.4 cm
   Answer: A. 8 - 0.3 x 3 = 7.1 cm. (Key point: Effective rain)

3. The 2-h, 7 cm storm starts at t = 3 h, so the flood is:
   A. 10 + 7.1 U3(t) + 6.4 U2(t-3)
   B. 7.1 U3 + 6.4 U2
   C. 10 + 8 U3 + 7 U2
   D. 10 + 7.1 U2(t) + 6.4 U3(t-3)
   Answer: A. Effective rains 7.1 and 6.4 cm with the second lagged by 3 h, plus 10 m^3/s base flow. (Key point: Superposition)

4. The flood peak is about:
   A. 2384 m^3/s at 10 h
   B. 1404 m^3/s at 6 h
   C. 896 m^3/s at 22 h
   D. 2120 m^3/s at 8 h
   Answer: A. From the table, the maximum is 2384.3 at t = 10 h. (Key point: Peak)

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