Question 7 of 9: Sizing 25 square footings from SPT data, and their bearing capacity

Question 7: Sizing 25 square footings from SPT data, and their bearing capacity (24 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.

Given.

Find. The plan dimension B of the square footings that limits settlement to 15 mm, the ultimate bearing capacity of that footing from Terzaghi's equation with the water table at the ground surface, and the professional advice that follows from comparing the two.

Approach. The column load is estimated from a stated service load per floor, the footing size is then found by equating the applied pressure to Meyerhof's allowable bearing pressure for the tolerable settlement (an iterative solve, because both the size factor and the averaged blow count depend on B), and Terzaghi's equation is applied afterwards with a friction angle derived from the overburden-corrected blow counts to show by how large a margin settlement rather than strength governs.

1. Estimate the column load. The exam gives no loading, and page-1 Note 7 invites the candidate to supply it. Taking a combined dead-plus-live service load of 10 kPa per floor — a conventional figure for a light commercial or residential frame — $$Q_{\text{total}}=5\ \text{floors}\times 600\ \text{m}^2\times 10\ \text{kPa}=30\,000\ \text{kN},\qquad Q_{\text{col}}=\frac{30\,000}{25}=\boxed{1200\ \text{kN per footing}}$$ Twenty-five footings on a 600 m² plan implies a 5 × 5 grid at roughly 4.9 m centres and a tributary area of 24 m², which is consistent with the assumed intensity.
2. Set up Meyerhof's settlement-based allowable pressure. For a square footing on sand with B greater than 1.22 m, the net allowable bearing pressure that produces a given elastic settlement is $$q_{\text{net(all)}}=11.98\,N_{60}\left(\frac{3.28B+1}{3.28B}\right)^{2}F_d\left(\frac{S_e}{25}\right),\qquad F_d=1+0.33\frac{D_f}{B}\le 1.33$$ with S_e in millimetres. The design blow count is the arithmetic mean of the corrected N₆₀ profile over the influence depth D_f to D_f + 2B.
3. Solve for B. The requirement is that the applied net pressure Q_col/B² equal the allowable value, so the equation is solved by iteration on B, re-averaging the blow counts at each cycle. Convergence gives B = 2.92 m with a mean blow count of 13.7 over 1.5 m to 7.35 m, and $$q_{\text{net(all)}}=11.98(13.7)\left(\frac{3.28(2.92)+1}{3.28(2.92)}\right)^{2}(1.169)(0.60)=140.3\ \text{kPa}=\frac{1200}{2.92^{2}}\ \checkmark$$ Rounding up to a constructible dimension, $$\boxed{B=3.0\ \text{m}\times 3.0\ \text{m square footings}}$$
4. Check the adopted size. At B = 3.0 m the influence zone runs to 7.5 m, the mean corrected blow count is 13.75, and $$F_d=1+0.33\frac{1.5}{3.0}=1.165,\qquad q_{\text{net(all)}}=11.98(13.75)(1.2136)(1.165)(0.60)=139.7\ \text{kPa}$$ against an applied pressure of 1200/3.0² = 133.3 kPa. Scaling settlement linearly with pressure, the predicted settlement is $$S_e=15\left(\frac{133.3}{139.7}\right)=\boxed{14.3\ \text{mm}<15\ \text{mm}\ \checkmark}$$
5. Derive a design friction angle from the blow counts. Bearing capacity needs φ′, which is obtained from the overburden-corrected values. With the water table at the surface, γ′ = 20 − 9.81 = 10.19 kN/m³, and with Liao and Whitman's correction C_N = (p_a/σ′_v)^0.5 capped at 1.7, the corrected profile is as follows.

   Depth (m)	σ′v (kPa)	CN	N60	(N1)60
   1.5	15.29	1.700 (capped)	10	17.0
   3.0	30.57	1.700 (capped)	12	20.4
   4.5	45.85	1.477	14	20.7
   6.0	61.14	1.279	16	20.5
   7.5	76.42	1.144	16	18.3
   Mean over the influence zone				19.4

6. Convert to a friction angle. Wolff's expression, fitted to the Peck–Hanson–Thornburn curve, gives $$\phi'=27.1+0.3(N_1)_{60}-0.00054(N_1)_{60}^{2}=27.1+0.3(19.4)-0.00054(19.4)^{2}=32.7^{\circ}$$ so φ′ = 33° is adopted. The corrected profile is essentially uniform at (N₁)₆₀ ≈ 19–21, which confirms that the rise in raw blow count with depth is an overburden effect and not a genuine increase in density.
7. Apply Terzaghi's equation with the water table at the surface. For a square footing on cohesionless soil, c′ = 0, and both the surcharge and the width terms must use the effective unit weight because the whole failure zone is submerged: $$q_u=1.3c'N_c+qN_q+0.4\gamma BN_{\gamma},\qquad q=\gamma' D_f=10.19(1.5)=15.29\ \text{kPa}$$ Terzaghi's factors at φ′ = 33° are N_q = 32.23 and N_γ = 31.94. Substituting, $$q_u=0+15.29(32.23)+0.4(10.19)(3.0)(31.94)=492.6+390.6=\boxed{883\ \text{kPa}}$$
8. Compare the two limits. Against the applied 133.3 kPa the factor of safety on bearing capacity is $$\text{FS}=\frac{883}{133.3}=\boxed{6.6}$$ and the allowable pressure at a conventional FS = 3 would be 883/3 = 294 kPa gross, or 289 kPa net — more than twice what settlement permits. Settlement, not bearing capacity, governs this foundation by a factor of well over two. The submergence is itself worth noting: with a deep water table the ultimate capacity would have been 1734 kPa, so the water table at ground level halves the strength of the foundation without changing the conclusion.

Comments to the owner

1. The design is settlement-controlled, and that is the number to watch. The footings are not close to a bearing failure; they are sized entirely by the 15 mm settlement limit. The owner should understand that a modest increase in floor loading, or a reduction in footing size to save concrete, translates almost directly into settlement rather than into any loss of safety against collapse.

2. The footings are large relative to the building, and a raft is the obvious alternative. Twenty-five footings at 3.0 m square occupy 225 m², which is 37.5 per cent of the 600 m² plan. Once individual footings cover more than about half the plan area the excavation, formwork and reinforcement of a mat foundation usually cost no more and perform better: a mat spreads the load to a greater depth, averages out the variability of the deposit, and gives far better control of differential settlement, which is what damages a structure. A comparative cost estimate for a mat should be prepared before the pad design is finalised.

3. Differential settlement, not total settlement, is the design criterion. For footings on sand, differential settlement is conventionally taken as up to about 75 per cent of the maximum total settlement, so a 15 mm total implies roughly 10 mm differential and an angular distortion near 10/4900 ≈ 1/490 at the assumed spacing. That is acceptable for a framed building, whose usual limit is 1/500 for cracking of panel walls, but it leaves little margin, and the tolerable settlement should be confirmed with the structural engineer and the architect rather than assumed.

4. The groundwater table at ground surface is the dominant construction issue. Excavating 1.5 m below a water table that stands at the surface requires dewatering by well points or sumps, with attendant risks of piping, boiling and loss of ground beneath adjacent footings, and settlement of neighbouring structures if the drawdown extends off site. Concrete must be placed in the dry. The basement or crawl space, if any, must be designed for hydrostatic uplift, and permanent drainage or a waterproof tanked structure will be needed. The water table also means that any future seismic assessment must address the liquefaction potential of a loose to medium-dense saturated sand: at (N₁)₆₀ ≈ 19 the sand is not obviously liquefiable, but this is a check the owner should commission explicitly if the site is in a seismically active region such as coastal British Columbia.

5. The investigation is thin for a five-storey building. A single blow-count profile has been used to size twenty-five footings across a 600 m² footprint. Sand deposits vary laterally, and the settlement prediction rests on a correlation with a scatter of roughly plus or minus fifty per cent. At least three or four boreholes distributed across the footprint, supplemented by cone penetration soundings for a continuous profile, would materially reduce the risk; plate load tests or a small-scale trial would reduce it further. The cost of that additional work is trivial beside the cost of remediating differential settlement in a completed building.

6. Options if the settlement proves unacceptable. In order of increasing cost: increase the footing size (settlement falls roughly in proportion to the applied pressure); found deeper, which increases F_d and mobilises higher blow counts; densify the upper sand by vibro-compaction or dynamic compaction, which raises N₆₀ from 10–14 into the low twenties and can halve the settlement; adopt a raft; or use short driven piles or rigid inclusions to transfer load below the influence zone.