---
title: "Intensity-duration graph from mass curve ordinates"
subject: "Engineering Hydrology"
chapter: 2
key: "semester_5/engineering_hydrology:intensity-duration-15-180min"
source: https://www.bctengineer.com/q/semester_5/engineering_hydrology/intensity-duration-15-180min
site: BCT Engineering
---

# Intensity-duration graph from mass curve ordinates

Engineering Hydrology · Chapter 2: Precipitation

- Asked in: 2073 Shrawan, 12 marks
- Video: [Hyetograph and Mass Curve - Hydrology - Civil Engineering - Engineers' Fate](https://www.youtube.com/watch?v=ZuoyXXAZfE4) — Engineers' Fate
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/engineering_hydrology/intensity-duration-15-180min
- Whole chapter: https://www.bctengineer.com/md/semester_5/engineering_hydrology/chapter-2

## Question

A storm commenced at 7:00 hours. The ordinates of the rainfall mass curve of this storm in mm as recorded by a recording rain gauge at 15 minute intervals are 0, 9.5, 17.0, 27.0, 40.5, 49.0, 63.0, 84.0, 95.0, 102.0, 110.0, 112.0 and 112.0. Plot the intensity duration graph by computing the maximum rainfall intensities for durations of 15, 30, 45, 60, 90, 120 and 180 minutes.

## Short answer

From the mass curve, maximum depths over each duration are converted to intensity. Intensities are 84.0, 70.0, 61.3, 57.0, 52.0, 47.5, and 37.3 mm/h for 15, 30, 45, 60, 90, 120, and 180 min, respectively.

## Answer

**Mass curve ordinates (mm)** at 15-min intervals from 07:00: 0, 9.5, 17.0, 27.0, 40.5, 49.0, 63.0, 84.0, 95.0, 102.0, 110.0, 112.0, 112.0.

Incremental rainfall (mm) per 15 min: 9.5, 7.5, 10.0, 13.5, 8.5, 14.0, 21.0, 11.0, 7.0, 8.0, 2.0, 0.

For each duration $D$, the maximum depth is the largest difference of mass curve ordinates $D$ minutes apart (found by sliding the window along the curve), and $i_{max} = \text{depth}/D$.

| Duration (min) | Max depth (mm) | Period (from 07:00) | Max intensity (mm/h) |
|---|---|---|---|
| 15 | 21.0 | 90-105 min | 84.0 |
| 30 | 35.0 | 75-105 min | 70.0 |
| 45 | 46.0 | 75-120 min | 61.3 |
| 60 | 57.0 | 45-105 min | 57.0 |
| 90 | 78.0 | 30-120 min | 52.0 |
| 120 | 95.0 | 0-120 min | 47.5 |
| 180 | 112.0 | 0-180 min | 37.3 |

Checks: 30 min = 14.0 + 21.0 = 35 mm; 1 h = 84.0 - 27.0 = 57 mm; 2 h = 95.0 - 0 = 95 mm.

**Intensity-duration graph:** plot duration on the x-axis and maximum intensity on the y-axis:

```text
 mm/h
 84 |*
 70 |   *
 61 |     *
 57 |       *
 52 |            *
 47.5|                *
 37 |                          *
    +---------------------------
     15 30 45 60  90 120   180 min
```

The intensity falls steadily as the duration increases.

**Answer: Maximum intensities = 84, 70, 61.3, 57, 52, 47.5 and 37.3 mm/h for 15, 30, 45, 60, 90, 120 and 180 min.**

## Check yourself

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

1. How are the 15-min incremental depths obtained from the mass curve?
   A. Differences of successive ordinates
   B. Sums of successive ordinates
   C. Ratios of successive ordinates
   D. Ordinates divided by 15
   Answer: A. Increment = difference of consecutive mass curve ordinates, e.g. 9.5, 7.5, 10.0 ... (Key point: Incremental rainfall)

2. What is the 15-min maximum intensity?
   A. 21 mm/h
   B. 84 mm/h
   C. 42 mm/h
   D. 112 mm/h
   Answer: B. Max 15-min depth is 21.0 mm (90-105 min); 21/0.25 = 84 mm/h. (Key point: 15-min maximum)

3. How is the maximum depth for D minutes found?
   A. Smallest difference of ordinates D apart
   B. D times the mean increment
   C. Largest difference of mass curve ordinates D minutes apart
   D. Total depth divided by D
   Answer: C. Slide a window of length D and take the largest ordinate difference. (Key point: Sliding window)

4. What is the 60-min maximum intensity and how?
   A. 84 mm/h
   B. 47.5 mm/h
   C. 112 mm/h
   D. 57 mm/h from 84.0 - 27.0
   Answer: D. Max 1-h depth = 84.0 - 27.0 = 57 mm in 45-105 min, i.e. 57 mm/h. (Key point: 1-hour maximum)

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