Elastic stresses in soil — Boussinesq solution for a point load A concentrated load of 700 kN acts at the ground surface on a thick clay layer (assume a semi-infinite, homogeneous, isotropic, elastic half-space). Using Boussinesq’s formula, estimate the vertical stress σ_v at depth z = 2 m and radial distance r = 1.0 m from the load application point.

Introduction / Context:
Boussinesq’s elastic solution is widely used in geotechnical engineering to estimate stress distribution caused by surface loads in a semi-infinite, homogeneous, isotropic, elastic half-space. This problem checks your ability to use the closed-form expression for the vertical stress beneath a surface point load at an arbitrary point (r, z).

Given Data / Assumptions:

• Point load P = 700 kN applied at the surface.
• Depth z = 2 m, radial distance r = 1 m.
• Soil mass idealized as elastic, homogeneous, isotropic, semi-infinite.
• We seek vertical stress σ_v at (r, z).

Concept / Approach:
The Boussinesq vertical stress beneath a surface point load is given by:
σ_v = (3 P z^3) / (2 π R^5)where R = (r^2 + z^2)^(1/2)Units consistency is crucial: if P is in kN and lengths in m, σ_v is obtained in kN/m^2, i.e., kPa.

Step-by-Step Solution:
1) Compute R = sqrt(r^2 + z^2) = sqrt(1^2 + 2^2) = sqrt(5) ≈ 2.236 m.2) Evaluate R^5 ≈ 2.236^5 ≈ 55.9.3) Numerator: 3 * P * z^3 = 3 * 700 * 8 = 16,800 (kN·m^3).4) Denominator: 2 * π * R^5 ≈ 2 * π * 55.9 ≈ 351.2.5) σ_v = 16,800 / 351.2 ≈ 47.83 kN/m^2 ≈ 47.8 kPa.

Verification / Alternative check:
Use the non-dimensional form σ_v = (3P / (2π z^2)) * 1 / (1 + (r/z)^2)^(5/2). With r/z = 0.5, (1 + 0.25)^(5/2) = 1.25^(2.5) ≈ 1.746, leading to the same value ≈ 47.8 kPa.

Why Other Options Are Wrong:
47.5, 47.6, and 47.7 kPa are close but result from rounding at intermediate steps; the precise calculation rounds to 47.8 kPa for two significant decimals.

Common Pitfalls:
Mixing units (using P in N rather than kN); forgetting that the formula yields kPa (kN/m^2) with P in kN and lengths in m; using R instead of R^5.

Final Answer:
47.8 kPa