---
title: "Approximate stress distribution methods"
subject: "Soil Mechanics"
chapter: 8
key: "semester_4/soil_mechanics:approximate-stress-distribution-methods"
source: https://www.bctengineer.com/q/semester_4/soil_mechanics/approximate-stress-distribution-methods
site: BCT Engineering
---

# Approximate stress distribution methods

Soil Mechanics · Chapter 8: Vertical Stresses Below Applied Loads

- Asked in: 2074 Bhadra, 3 marks; 2078 Poush (asked 2 times)
- Video: [Approximate method (2:1 Method) to determine vertical stress due to applied loads](https://www.youtube.com/watch?v=QrdMOhcB3cQ) — Civil@VRS
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/soil_mechanics/approximate-stress-distribution-methods
- Whole chapter: https://www.bctengineer.com/md/semester_4/soil_mechanics/chapter-8

## Question

Describe the approximate stress distribution methods for loaded areas.

## Short answer

Approximate methods assume load spreads linearly with depth. The 2:1 (vertical:horizontal) method gives average stress Δσz = Q/[(B+z)(L+z)]; the 60° method spreads at 30° from vertical. These yield quick estimates useful for settlement calculations at depths around 1–4B.

## Answer

Approximate methods give a quick estimate of the vertical stress below a loaded area without using elastic theory. They assume that the load spreads out with depth over an increasing area.

#### 2:1 method (2 vertical : 1 horizontal)

The load $Q$ is assumed to spread at a slope of 2 vertical to 1 horizontal (about 26.6° from the vertical) on all sides. At depth $z$, the loaded area increases by $z$ in each direction.

```text
     |<-- B -->|
     ▇▇▇▇▇▇▇▇▇▇▇          ground
      \       /
       \     /     2V : 1H
        \   /
   z     \ /
      |<- B+z ->|
```

- Rectangular footing $B\times L$:

$$
\Delta\sigma_z = \frac{Q}{(B+z)(L+z)}
$$

- Square footing ($B\times B$): $\Delta\sigma_z = \dfrac{Q}{(B+z)^2}$
- Strip footing (per unit length): $\Delta\sigma_z = \dfrac{Q}{B+z}$
- Circular footing of diameter $D$: $\Delta\sigma_z = \dfrac{Q}{\frac{\pi}{4}(D+z)^2}$

#### 60° distribution (30° method)

The load is assumed to spread at an angle of $30^\circ$ with the vertical (60° with horizontal) on each side, so the width at depth $z$ is $B + 2z\tan30^\circ$:

$$
\Delta\sigma_z = \frac{Q}{(B+2z\tan30^\circ)(L+2z\tan30^\circ)}
$$

#### Comments

- The result is the average stress over the width at depth $z$, not the maximum under the centre.
- The 2:1 method agrees reasonably with Boussinesq (under the centre) at depths of about 1 to 4 times the width of the footing. It is useful for quick settlement estimates of layered soils.

## Check yourself

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

1. In the 2:1 method, load at depth z spreads over:
   A. (B+z)(L+z)
   B. (B+2z)(L+2z)
   C. B L
   D. (B+z/2)(L+z/2)
   Answer: A. The width increases by z on each side in total, giving (B+z)(L+z). (Key point: 2:1 method)

2. Stress for a strip footing by 2:1 is:
   A. Q/(B+z)
   B. Q/(B+z)^2
   C. Q/B
   D. Q/(2z)
   Answer: A. Per unit length width is B+z. (Key point: Strip footing)

3. In the 30 degree method, width at depth z is:
   A. B + 2z tan30
   B. B + z
   C. B + z tan60
   D. B + 2z
   Answer: A. The spread is at 30 degrees from vertical on each side. (Key point: 60 degree method)

4. The computed stress is:
   A. Maximum at centre
   B. Average over the spread width at depth z
   C. Zero at edges
   D. Equal to Boussinesq always
   Answer: B. It is an average value, not the peak. (Key point: Nature of result)

## Related in Soil Mechanics

- [Boussinesq stress below circle center](https://www.bctengineer.com/question.md?key=semester_4%2Fsoil_mechanics%3Aboussinesq-circle-center-stress) — Soil Mechanics, chapter 8
- [Ring foundation 3.60 m stress](https://www.bctengineer.com/question.md?key=semester_4%2Fsoil_mechanics%3Aring-foundation-3-6-m-stress) — Soil Mechanics, chapter 8
- [T-shaped foundation, stress at P](https://www.bctengineer.com/question.md?key=semester_4%2Fsoil_mechanics%3At-shaped-foundation-stress-p) — Soil Mechanics, chapter 8
- [Annular ring, stress at five depths](https://www.bctengineer.com/question.md?key=semester_4%2Fsoil_mechanics%3Aannular-ring-stress-with-depth) — Soil Mechanics, chapter 8

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