Boussinesq Solution – Vertical Stress on the Axis Beneath a Point Load For a point load Q acting at the ground surface of a homogeneous, isotropic, elastic half-space, the vertical stress at depth z directly below the load (on the axis of loading) is given by which expression?

Introduction / Context:
The Boussinesq elastic solution provides closed-form expressions for stress distribution in a semi-infinite, homogeneous, isotropic, linear-elastic half-space due to a surface point load. The axial stress beneath the load is frequently used in settlement and stress distribution calculations below footings and embankments.

Given Data / Assumptions:

• Point load Q applied at the surface.
• Elastic, homogeneous, isotropic half-space.
• Interest is the vertical stress σ_z at a depth z directly beneath the load (r = 0 along the axis).

Concept / Approach:

From Boussinesq’s equations, the vertical stress at a general point is σ_z = (3 Q / (2 π)) * (z^3 / R^5), where R = (r^2 + z^2)^(1/2) and r is radial distance from the load axis. On the axis (r = 0), R = z, so σ_z simplifies to σ_z = (3 Q) / (2 π z^2). This elegant result shows that the axial vertical stress decays with the square of depth and is independent of Poisson’s ratio for the axial point.

Step-by-Step Solution:

Verification / Alternative check:

The same expression can be recovered by integrating the axisymmetric stress influence factor I_z = 3/(2π) * 1/(1 + (r/z)^2)^(5/2) at r = 0, which equals 3/(2π).

Why Other Options Are Wrong:

(a), (d), and (e) have incorrect coefficients. (c) writes a general form that reduces to (b) only when the correct power of R is used; with R^3 the units are wrong.

Common Pitfalls:

Forgetting that on-axis stress is independent of Poisson’s ratio; mixing up depth dependence (1/z^2) with off-axis expressions.

Final Answer:

σ_z = (3 Q) / (2 π z^2)