---
title: "Ring footing 8 m / 4 m, stress at 8 m"
subject: "Soil Mechanics"
chapter: 8
key: "semester_4/soil_mechanics:ring-footing-8m-depth-stress"
source: https://www.bctengineer.com/q/semester_4/soil_mechanics/ring-footing-8m-depth-stress
site: BCT Engineering
---

# Ring footing 8 m / 4 m, stress at 8 m

Soil Mechanics · Chapter 8: Vertical Stresses Below Applied Loads

- Asked in: 2075 Bhadra, 6 marks
- Video: [Numerical Example on determination of Vertical stress due to 'ring footing'](https://www.youtube.com/watch?v=ZCyLefo8OVY) — 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/ring-footing-8m-depth-stress
- Whole chapter: https://www.bctengineer.com/md/semester_4/soil_mechanics/chapter-8

## Question

A ring footing of external diameter 8 m and internal diameter 4 m rests at a depth 2 m below the ground surface. It carries a load intensity of 150 kN/m². Find the vertical stress at a depth of 8 m along the axis of the footing below the footing base. Neglect the effect of the excavation on the stress.

## Short answer

Stress = q[1/(1+(r/z)²)^(3/2) − 1/(1+(R/z)²)^(3/2)]. For R=4 m, r=2 m, z=8 m, q=150 kN/m²: outer factor=0.2845, inner=0.0869, net=0.1976 × 150 = 29.6 kN/m².

## Answer

#### Given

Outer $D = 8$ m ($R = 4$ m), inner $d = 4$ m ($r = 2$ m), $q = 150$ kN/m². Depth of interest is 8 m below the base of the footing, so $z = 8$ m. The effect of excavation is neglected.

#### Calculation

$$
\Delta\sigma_z = q\left[\left(1 - \frac{1}{(1+(R/z)^2)^{3/2}}\right) - \left(1 - \frac{1}{(1+(r/z)^2)^{3/2}}\right)\right]
$$

- Outer: $R/z = 0.5$, $(1.25)^{1.5} = 1.3975$, factor $= 1 - 0.7155 = 0.2845$
- Inner: $r/z = 0.25$, $(1.0625)^{1.5} = 1.0952$, factor $= 1 - 0.9131 = 0.0869$

$$
\Delta\sigma_z = 150\,(0.2845 - 0.0869) = 150\times0.1976 = 29.6\ \text{kN/m}^2
$$

**Answer: Vertical stress at 8 m below the footing base along the axis $\approx 29.6$ kN/m².**

## Check yourself

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

1. The footing base is 2 m below ground and the point is 8 m below the footing base. Which z is used?
   A. 10 m
   B. 6 m
   C. 2 m
   D. 8 m
   Answer: D. Stress is measured from the footing base, so z = 8 m; the excavation effect is neglected. (Key point: Depth z)

2. With R = 4 m, r = 2 m and z = 8 m, the ratios R/z and r/z are
   A. 0.5 and 0.25
   B. 2 and 4
   C. 0.25 and 0.5
   D. 1 and 0.5
   Answer: A. 4/8 = 0.5 and 2/8 = 0.25. (Key point: Ratios)

3. The axial stress for q = 150 kN/m2 is about
   A. 42.7 kN/m2
   B. 29.6 kN/m2
   C. 13.0 kN/m2
   D. 150 kN/m2
   Answer: B. 150 x (0.2845 - 0.0869) = 150 x 0.1976 = 29.6 kN/m2. (Key point: Result)

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