24-Bld-A6 Geotechnical Materials and Analysis, December 2017, Q5

Question 5 (6 + 14 = 20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

(i) Assumptions of elastic (Boussinesq-type) stress theory: the soil is treated as a semi-infinite, homogeneous, isotropic, linearly elastic half-space; it is assumed weightless (only the stress INCREMENT due to the applied load is computed, superposed on the existing in-situ stress); the load acts at/on the surface of the half-space with no other loads present; and the material never yields — stress and strain remain proportional at every load level.

Limitations: real soil is not linearly elastic (stiffness changes with confining stress and strain level, especially as failure is approached); natural deposits are layered and generally get stiffer with depth, not homogeneous; many soils (particularly clays) are anisotropic (different stiffness vertically vs. horizontally), contrary to the isotropic assumption; the theory ignores a rigid layer at finite depth (a real shallow rigid stratum concentrates stress more than the idealised infinite half-space predicts); and it takes no account of footing size, rigidity, or embedment beyond the loaded area itself. Despite these idealisations, field measurements generally agree with Boussinesq-based predictions well enough for routine settlement and stress-distribution design, which is why the method remains standard practice.

Typical variation of vertical stress with depth (left) and with horizontal distance from the load axis at three depths (right), due to a point load — the classic Boussinesq "pressure bulb": narrower and taller near the surface, wider and flatter with depth.

(ii) Given. The loaded area in Figure 2 is a 7 m × 4 m rectangle (5 m + 2 m wide, 1 m + 3 m deep), loaded at q = 100 kPa; point A sits at the internal division of these dimensions — 5 m from one side and 2 m from the other (width-wise), 1 m from one edge and 3 m from the other (length-wise) — so A is a shared corner of four sub-rectangles that together tile the whole loaded area.

Figure 2 — rectangular loaded area, split into four sub-rectangles sharing corner A, for superposition.

Find. The increase in vertical stress $\Delta\sigma_z$ at 2.0 m depth below A, by two methods, and a comparison.

Approach. Method 1 (exact): since A is not a corner of the actual loaded rectangle, decompose the area into four rectangles that DO share a corner at A and sum the exact Boussinesq corner-influence factors from the formula sheet. Method 2 (approximate): the formula sheet's approximate formula $\sigma_z=qBL/[(B+z)(L+z)]$ is written for the CENTRE of a B×L rectangle; mirroring the loaded rectangle about corner A produces a $2B\times2L$ rectangle centred exactly on A, so the corner stress equals one quarter of that larger rectangle's centre-formula stress — giving $\sigma_{z,corner}\approx qBL/[(2B+z)(2L+z)]$, applied to each of the same four sub-rectangles.

Method 1 — exact corner superposition. With $z=2$ m, the influence factor $I(m,n)$ (formula sheet, $m=B/z,\,n=L/z$) evaluated for each sub-rectangle: $$I(2.5,0.5)=0.136,\quad I(1,0.5)=0.120,\quad I(2.5,1.5)=0.227,\quad I(1,1.5)=0.194,$$ $$\Delta\sigma_z = q\sum I = 100(0.136+0.120+0.227+0.194) = 100(0.677) \boxed{=\ 67.7\ \text{kPa}}.$$
Method 2 — approximate (2:1-type) corner formula. Applying $\sigma_{z,corner}=qBL/[(2B+z)(2L+z)]$ to the same four sub-rectangles: $$\underbrace{10.4}_{5\times1}+\underbrace{8.3}_{2\times1}+\underbrace{15.6}_{5\times3}+\underbrace{12.5}_{2\times3} \boxed{=\ 46.9\ \text{kPa}}.$$
Compare. The two methods differ by about 30% (67.7 kPa exact vs. 46.9 kPa approximate). The approximate method assumes the load spreads uniformly at a fixed 2(vertical):1(horizontal) slope from the loaded edges in every direction, which understates the stress concentration directly beneath a point close to the centre of mass of the loaded area (as A effectively is here, sitting well inside the 7×4 m footprint) — the exact Boussinesq solution captures that concentration more faithfully. The approximate method's convenience (a single closed-form calculation with no charts or superposition of transcendental influence factors) is traded for reduced accuracy near heavily loaded interior points; it is best reserved for quick preliminary estimates or for points once a filled-in area is small.

Question 5(ii) — final results
Quantity	Value
Method 1 (exact Boussinesq superposition)	67.7 kPa
Method 2 (approximate, quartered 2:1-type)	46.9 kPa
Difference	≈ 31% (approximate method lower)

Concept & Theory

Topic. Vertical stress beneath an interior point of a rectangular loaded area, by exact and approximate superposition.

Key relations. Exact rectangular-corner influence factor I(m,n) (Boussinesq); approximate centre formula $\sigma_z=qBL/[(B+z)(L+z)]$, extended to a corner via the mirror/quartering identity $\sigma_{z,corner}(B,L,z)=\tfrac14\sigma_{z,centre}(2B,2L,z)$.

Why this works. Both stress formulas are linear in the applied pressure, so any point — corner, edge, or interior — can be built from rectangles sharing a corner at that point; a point inside the loaded area (like A here) is treated as the shared corner of the four panels the point divides the area into.

Common pitfall. Applying the approximate centre formula directly to the whole 7×4 m rectangle when the point of interest is NOT at its centre; forgetting that m and n are interchangeable but each sub-rectangle needs its own m, n pair.

Source. Das, Principles of Geotechnical Engineering, Ch. 8 (stress distribution); exam formula sheet, rectangular-loading equations.

Question 5 of 7

Question 5 (6 + 14 = 20 marks)