Question 6 of 9: Footing size from the SPT profile, and Terzaghi bearing capacity

Question 6: Footing size from the SPT profile, and Terzaghi 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 size $B$ of the square footings such that the settlement of each footing does not exceed 15 mm; then the ultimate and allowable bearing capacity of that footing from Terzaghi's equation with the water table at ground level; and finally the professional comments that should accompany the design.

Approach. Convert the storey allowance into a column load, then size the footing by trial from Meyerhof's settlement relation (as modified by Bowles) using the mean $N_{60}$ over the depth of influence $D_f$ to $D_f + 2B$, and finally check the adopted size against Terzaghi's bearing-capacity equation with the buoyant unit weight, since the water table stands at the surface.

1. Convert the storey allowance into the design load on one footing. The hint fixes the total service pressure over the footprint as five storeys at 10 kPa each: $$Q_{total} = 5(10\ \text{kPa})(600\ \text{m}^2) = 30\,000\ \text{kN}, \quad Q = \frac{30\,000}{25} = 1200\ \text{kN}$$ so each of the 25 columns delivers $Q = 1200\ \text{kN}$ to its own footing.
2. Choose the empirical settlement method and its influence depth. Meyerhof's SPT relation, in the form modified by Bowles and reproduced by Das, gives the net pressure that produces a stated elastic settlement in sand: $$q_{net} = \frac{N_{60}}{0.08}\left(\frac{B+0.3}{B}\right)^{2} F_d \left(\frac{S_e}{25}\right) \quad (B \gt 1.22\ \text{m})$$ with $F_d = 1 + 0.33(D_f/B) \le 1.33$, $S_e$ in millimetres and $q_{net}$ in kPa. The $N_{60}$ to be used is the mean over the zone of significant stress increase, taken from founding level to a depth of $2B$ below the base.
3. Iterate on $B$. The applied net pressure $Q/B^2$ falls as $B^2$ while the permissible pressure falls only slowly, so a single crossing exists. Trials with the mean $N_{60}$ recomputed for each influence depth give:

   Trial footing sizes, tolerable settlement 15 mm

   $B$ (m)	Influence zone (m)	Mean $N_{60}$	$F_d$	$q_{net,\,all}$ (kPa)	Applied $Q/B^2$ (kPa)	Verdict
   2.5	1.5–6.5	13.0	1.198	146.5	192.0	fails
   2.7	1.5–6.9	13.0	1.183	142.4	164.6	fails
   2.9	1.5–7.3	13.0	1.171	139.0	142.7	fails, marginally
   3.0	1.5–7.5	13.6	1.165	143.8	133.3	satisfactory

   The step at $B = 3.0\ \text{m}$ is real rather than numerical: the influence zone reaches the 7.5 m sampling depth, where $N_{60} = 16$, and the mean rises from 13.0 to 13.6.
4. Confirm the adopted size. For $B = 3.0\ \text{m}$ the depth factor is $F_d = 1 + 0.33(1.5/3.0) = 1.165$ and the mean corrected blow count over 1.5 m to 7.5 m is $N_{60} = (10+12+14+16+16)/5 = 13.6$, so $$q_{net,\,all} = \frac{13.6}{0.08}\left(\frac{3.3}{3.0}\right)^{2}(1.165)\left(\frac{15}{25}\right) = 143.8\ \text{kPa}$$ against an applied pressure of $1200/3.0^2 = 133.3\ \text{kPa}$. Comparing the column pressure $Q/B^2$ directly with the net allowable pressure is legitimate here because the weight of the footing and of the backfill over it roughly replaces the weight of the soil excavated to 1.5 m, so $Q/B^2$ is effectively the net pressure increase at founding level (subtracting the effective overburden as well would only add margin). The settlement criterion is met with about eight per cent in hand, so $$\boxed{B = 3.0\ \text{m square footings, } 3.0 \times 3.0\ \text{m, at } D_f = 1.5\ \text{m}}$$
5. Set up Terzaghi's equation with the water table at the surface. The sand is cohesionless, so $c' = 0$ and the cohesion term vanishes. With the water table at ground level the effective unit weight applies both to the surcharge term and to the $N_\gamma$ term: $$\gamma' = \gamma_{sat} - \gamma_w = 20 - 9.81 = 10.19\ \text{kN/m}^3, \qquad q = \gamma' D_f = 10.19(1.5) = 15.29\ \text{kPa}$$ For a square footing Terzaghi's equation is $q_u = 1.3c'N_c + qN_q + 0.4\gamma' BN_\gamma$.
6. Select the friction angle and read the bearing-capacity factors. The mean corrected blow count in the bearing zone is $N_{60} \approx 14$. Wolff's correlation $\phi' = 27.1 + 0.3N_{60} - 0.00054N_{60}^2$ gives $31.2^\circ$ and the Peck, Hanson and Thornburn chart gives about $32^\circ$ for the same blow count; adopt $\phi' = 32^\circ$, a medium-dense sand. Terzaghi's factors at $32^\circ$ are $N_q = 28.52$, $N_c = 44.04$ and $N_\gamma = 26.87$.

   Check: the friction angle is inferred from a blow-count correlation, not measured. A cone sounding at the site would settle it directly; if $\phi'$ proved to be $30^\circ$ rather than $32^\circ$ the ultimate capacity would fall by about a quarter, which the factor of safety below absorbs comfortably.

7. Evaluate the ultimate bearing capacity. Substituting into Terzaghi's equation, $$q_u = (15.29)(28.52) + 0.4(10.19)(3.0)(26.87) = 436.0 + 328.5 = 764.4\ \text{kPa}$$ and the net ultimate value, which is what a factor of safety is applied to, is $q_{u(net)} = 764.4 - 15.29 = 749.2\ \text{kPa}$. At a factor of safety of 3, $$\boxed{q_{all(net)} = \frac{749.2}{3} = 249.7\ \text{kPa}, \qquad q_{all(gross)} = 265.0\ \text{kPa}}$$
8. Compare the two criteria. The net applied pressure is $133.3 - 15.29 = 118.0\ \text{kPa}$, so the true factor of safety against a bearing-capacity failure is $$FS_{bearing} = \frac{749.2}{118.0} = 6.35$$ The footing is therefore twice as strong as it needs to be against shear and is sized entirely by the 15 mm settlement limit — exactly the situation Question 4 describes in the abstract.

Comments to the owner, as consultant. Four things need saying. First, the design is a settlement design, not a strength design: the footings are governed by the 15 mm tolerable movement, and a factor of safety of 6.35 against shear failure is what falls out of that, not a target. If the structure can in fact tolerate the more usual 25 mm, the footings could shrink to 2.3 m square and the substructure cost would fall materially — that is a question for the structural engineer and it is worth asking before the design is fixed. Second, the footings are close to raft territory: 25 footings at 3.0 m square occupy 225 m² of the 600 m² footprint, or 37.5 per cent. A raft should be priced against the pad-footing scheme, and it would also solve the water problem described next. Third, the water table at ground level dominates the site. It halves the effective unit weight and so roughly halves the $N_\gamma$ contribution; more practically, every excavation to 1.5 m will need dewatering or a sump, the formation will be prone to boiling and softening, and the concrete must be placed against a stable base. Loose saturated sand with $N_{60} = 10$ at founding level is also a liquefaction candidate, and in the seismic zones of coastal British Columbia a triggering assessment under the NBCC design earthquake is not optional. Fourth, the ground investigation is thin for the decision being made: a single blow-count profile supports neither an assessment of lateral variability across a 600 m² footprint nor a liquefaction screening. Cone soundings at three or four locations, with a few sampled boreholes for gradation, would remove most of the residual uncertainty at a cost that is trivial beside the substructure.