---
title: "Draw isobar diagram of 0.1Q"
subject: "Soil Mechanics"
chapter: 8
key: "semester_4/soil_mechanics:isobar-diagram-point-one-q"
source: https://www.bctengineer.com/q/semester_4/soil_mechanics/isobar-diagram-point-one-q
site: BCT Engineering
---

# Draw isobar diagram of 0.1Q

Soil Mechanics · Chapter 8: Vertical Stresses Below Applied Loads

- Asked in: 2075 Baisakh, 5 marks
- Video: [Isobar diagram | Pressure bulb Diagram | for 0.1Q | Soil Mechanics | Chapter 8](https://www.youtube.com/watch?v=h9yvHBhICdw) — Sudip Dhakal
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/soil_mechanics/isobar-diagram-point-one-q
- Whole chapter: https://www.bctengineer.com/md/semester_4/soil_mechanics/chapter-8

## Question

Draw the isobar diagram of 0.1Q.

## Short answer

The 0.1Q isobar for a point load is obtained by solving Boussinesq's equation for Δσz = 0.1Q, yielding coordinates such as maximum depth z = 2.19 m on the axis and maximum radius r ≈ 0.94 m at z ≈ 1.18 m. The closed bulb begins at the load point, with stress exceeding 0.1Q inside.

## Answer

The isobar of $0.1Q$ is the contour of points in the soil below a point load $Q$ where the vertical stress equals $0.1Q$ (per m², with $Q$ in kN and distances in m). Use Boussinesq's equation:

$$
\Delta\sigma_z = \frac{3Q}{2\pi z^2}\left[\frac{1}{1+(r/z)^2}\right]^{5/2} = \frac{Q}{z^2}\,I_B
$$

with $I_B = 0.4775\,[1+(r/z)^2]^{-5/2}$.

#### Computation

Setting $\Delta\sigma_z = 0.1\,Q$: $z^2 = I_B/0.1$, so $z = \sqrt{10\,I_B}$ and $r = (r/z)\,z$.

| $r/z$ | $I_B$ | $z$ (m) | $r$ (m) |
|---|---|---|---|
| 0 | 0.4775 | 2.185 | 0 |
| 0.2 | 0.4329 | 2.081 | 0.416 |
| 0.4 | 0.3295 | 1.815 | 0.726 |
| 0.6 | 0.2214 | 1.488 | 0.893 |
| 0.8 | 0.1386 | 1.177 | 0.942 |
| 1.0 | 0.0844 | 0.919 | 0.919 |
| 1.2 | 0.0513 | 0.717 | 0.860 |
| 1.5 | 0.0251 | 0.501 | 0.751 |
| 2.0 | 0.0085 | 0.292 | 0.585 |

#### Diagram

Plot the points $(\pm r, z)$ with $z$ downward and join them by a smooth curve.

```text
        Q (load)
  ------●-------------  surface
     .  |  .       r max = 0.94 m at z = 1.18 m
    .   |   .
    .   |   .
     .  |  .
      . | .
        ●    z max = 2.19 m on the axis (r = 0)
```

The isobar is a closed bulb that begins and ends at the load point on the surface and extends to $z = 2.19$ m on the axis. Inside the bulb the stress is greater than $0.1Q$.

## Check yourself

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

1. Which equation is used to draw the 0.1Q isobar below a point load?
   A. Terzaghi's bearing capacity equation
   B. Boussinesq's equation, delta_sigma_z = (Q/z^2) I_B
   C. Westergaard's equation
   D. The 2:1 method only
   Answer: B. Boussinesq's equation with I_B = 0.4775[1+(r/z)^2]^(-5/2) gives the vertical stress below a point load. (Key point: Equation used)

2. Setting delta_sigma_z = 0.1Q, how is z obtained for a chosen r/z?
   A. z = 0.1 I_B
   B. z = sqrt(10 I_B)
   C. z = 10 I_B
   D. z = I_B^2
   Answer: B. Q I_B / z^2 = 0.1 Q gives z^2 = I_B/0.1 = 10 I_B, so z = sqrt(10 I_B). (Key point: Solving for depth)

3. At r = 0 (on the axis) what is the depth of the 0.1Q isobar?
   A. 0.92 m
   B. 1.18 m
   C. 2.19 m
   D. 4.77 m
   Answer: C. With I_B = 0.4775, z = sqrt(4.775) = 2.185 m, about 2.19 m. (Key point: Axis depth)

4. Which statement about the 0.1Q isobar is correct?
   A. Stress inside the bulb is less than 0.1Q
   B. It is a straight line
   C. It extends infinitely deep
   D. It is a closed bulb; stress inside is greater than 0.1Q
   Answer: D. The isobar is a closed bulb starting and ending at the load point, enclosing soil where stress exceeds 0.1Q. (Key point: Shape and meaning)

## Related in Soil Mechanics

- [What is an isobar diagram](https://www.bctengineer.com/question.md?key=semester_4%2Fsoil_mechanics%3Aisobar-diagram-define) — Soil Mechanics, chapter 8