Elasticity under a surface point load: directly beneath a point load P at depth z in a homogeneous, isotropic, elastic half-space, the vertical stress σz can be expressed as σz = k * P / z^2. Which exact expression is correct?

Introduction / Context:
Stress distribution in soils from concentrated surface loads is often estimated using Boussinesq's elastic solution for a semi-infinite half-space. Directly under a point load, the formula simplifies to a closed form that decays with the square of depth, a key tool in foundation settlement and stress bulb calculations.

Given Data / Assumptions:

• Homogeneous, isotropic, linear elastic half-space.
• Surface point load P applied at r = 0.
• Interest in vertical stress σz along the load axis at depth z.

Concept / Approach:
The general Boussinesq expression for σz at radial distance r is σz = (3 P / 2 π) * (z^3) / (r^2 + z^2)^(5/2). Setting r = 0 (directly below the load) reduces the expression to σz = (3 P) / (2 π z^2). This highlights the 1/z^2 decay with depth and the proportionality constant k = 3/(2π).

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