The vertical stress σz under a uniformly distributed circular load based on Boussinesq’s theory is (where p is surface pressure, z is depth at which σz is computed and a is radius of loaded area.)
- Ap [1 − z³ / (a² + z²)3/2]Correct
- Bp [1 + z³ / (a² + z²)3/2]
- Cp [1 − z² / (a² + z²)3/2]
- Dp [1 + z² / (a² + z²)3/2]
Solution
Concept
Vertical stress under the centre of a circular load
Formula
σz = p [1 − z³ / (a² + z²)3/2]
Step-by-step solution
1. Integrating Boussinesq's point-load solution over the circular area gives σz = p [1 − z³/(a² + z²)3/2] below the centre. 2. Check: at z = 0, σz = p; as z → ∞, σz → 0.
Explanation
The formula gives the full surface pressure at the surface and a vanishing stress at great depth, which confirms the sign and the z³ term.
Why the other options are wrong
Options with a plus sign give a stress larger than p. The z² form has the wrong dimension.
Shortcut
σz = p[1 − (1 + a²/z²)−3/2].
Common mistake
Using a plus sign.
Correct answer: A) p [1 − z³ / (a² + z²)^(3/2)]