Question 6 of 9: Sizing 25 square pads on sand from SPT data, then checking Terzaghi

Question 6: Sizing 25 square pads on sand from SPT data, then checking Terzaghi (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. (a) the plan size $B$ of one square pad such that the settlement does not exceed 15 mm; (b) the ultimate and allowable bearing capacity of that pad from Terzaghi’s equation with the water table at the surface; and (c) the advice a consultant should give the owner, together with the assumptions made.

Approach. On sand, settlement almost always governs, so size the pad with an empirical settlement method driven by $N_{60}$ — the modified Meyerhof equation — then verify the result against Terzaghi’s shear criterion with the submerged unit weights that the stated water table demands.

1. Convert the storey allowance into a column load. The hint gives a uniform 10 kPa per storey over the plan area, so the whole structure delivers $$\begin{aligned}Q_{total}&=5\times 10\ \text{kPa}\times 600\ \text{m}^2=30\,000\ \text{kN} \\ Q_{col}&=\frac{30\,000}{25}=\boxed{1200\ \text{kN per column}}\end{aligned}$$ Twenty-five columns on a 600 m² plan implies a square grid of $\sqrt{600/25}=4.90\ \text{m}$, which is worth noting now because it will limit how large the pads may become.
2. Choose the design method and defend it. The question asks for a suitable empirical design method. On a granular deposit characterised by SPT, the appropriate method is the modified Meyerhof settlement equation, which returns the net pressure that produces a stated settlement directly: $$\begin{aligned}q_{net(all)}&=11.98\,N_{60}\left(\frac{3.28B+1}{3.28B}\right)^{2}F_d\,\frac{S_e}{25.4}\ \text{kPa} \\ &(B \gt 1.22\ \text{m})\end{aligned}$$ with the depth factor $F_d=1+0.33\,D_f/B\le 1.33$ and $S_e$ in millimetres. Because $B$ appears on both sides, the size is found by iteration.
3. Take the representative $N_{60}$ over the depth the pad will stress. The stress bulb of a square footing is spent by about $2B$ below founding level, so for a trial $B=3.0\ \text{m}$ the relevant interval is 1.5 m to 7.5 m, covering the readings 10, 12, 14, 16 and 16: $$\bar{N}_{60}=\frac{10+12+14+16+16}{5}=13.6$$ This is a loose-to-medium-dense sand, and the profile is still gaining density with depth over the influence zone, which is favourable.
4. Iterate on B. With $\bar{N}_{60}=13.6$, $D_f=1.5\ \text{m}$ and $S_e=15\ \text{mm}$, at $B=3.0\ \text{m}$ the depth factor is $F_d=1+0.33(1.5/3.0)=1.165$ and the width term is $\left(\tfrac{3.28(3)+1}{3.28(3)}\right)^{2}=1.214$, so $$q_{net(all)}=11.98(13.6)(1.214)(1.165)(15/25.4)=136.0\ \text{kPa}$$ while the applied net pressure is $q=1200/3.0^{2}=133.3\ \text{kPa}$. The pad is adequate with 2 per cent to spare. Repeating at $B=2.9\ \text{m}$ gives $q_{net(all)}=131.5\ \text{kPa}$ against an applied 142.7 kPa, which fails, so 3.0 m is the minimum size on a 0.1 m increment.
5. Adopt the footing. $$\boxed{B=3.0\ \text{m square pads},\ D_f=1.5\ \text{m},\ q_{applied}=133\ \text{kPa}\ \lt\ q_{net(all)}=136\ \text{kPa}}$$

With the size fixed, the second part of the question asks for the bearing capacity of that same pad from Terzaghi’s equation. Because the water table is at the natural ground surface, both the surcharge term and the self-weight term must be computed on effective, that is submerged, unit weights.

6. Friction angle from the blow count. Terzaghi’s factors need $\phi'$, which is not given, so it is taken from the corrected blow count with the Wolff (1989) correlation quoted in Das, $\phi'=27.1+0.3N_{60}-0.00054N_{60}^{2}$: $$\begin{aligned}\phi'&=27.1+0.3(13.6)-0.00054(13.6)^{2} \\ &=31.1^{\circ}\ \Rightarrow\ \phi' \\ &=31^{\circ}\ \text{adopted}\end{aligned}$$
7. Terzaghi bearing-capacity factors at 31 degrees. $$\begin{aligned}N_q&=\frac{e^{(3\pi/2-\phi')\tan\phi'}}{2\cos^{2}(45^{\circ}+\phi'/2)}=25.28 \\ N_c&=(N_q-1)\cot\phi'=40.41\end{aligned}$$ and $N_\gamma=22.40$ from Terzaghi’s tabulated values (Das, Table 3.1, after Kumbhojkar 1993). Naming the source of the chart value is required by Note 6 on the cover page.
8. Effective unit weight and surcharge with the water table at the surface. $$\begin{aligned}\gamma'&=\gamma_{sat}-\gamma_w=20-9.81=10.19\ \text{kN}/\text{m}^3 \\ q&=\gamma' D_f=10.19(1.5)=15.29\ \text{kPa}\end{aligned}$$ The water table at the surface roughly halves both terms relative to a dry profile, which is the single most punishing assumption in this question.
9. Terzaghi’s equation for a square footing. With $c=0$ for a clean sand, $$q_u=1.3cN_c+qN_q+0.4\gamma' BN_\gamma=0+15.29(25.28)+0.4(10.19)(3.0)(22.40)$$ $$q_u=386.4+273.9=\boxed{660\ \text{kPa gross}}$$ and the net ultimate value is $q_{nu}=660.3-15.29=645\ \text{kPa}$.
10. Allowable value and the margin actually carried. At a factor of safety of 3 the net allowable pressure from shear is $q_{net(all)}=645/3=215\ \text{kPa}$, against the 133 kPa the columns apply, so the true factor of safety against a bearing failure is $$\mathrm{FS}=\frac{645}{133.3}=\boxed{4.8}$$ Shear failure is a long way from being the problem: the settlement criterion, which permitted only 136 kPa, is what sized the pad.

Comments to the owner, as the consultant. Six points are worth putting in writing.

First, the scheme is on the boundary at which a raft becomes the better buy. Twenty-five 3.0 m pads occupy 225 m² of the 600 m² footprint, 37.5 per cent, on a 4.90 m grid, leaving only 1.9 m clear between adjacent pads. At that spacing the stress bulbs of neighbouring footings overlap, so the real settlement will exceed the isolated-footing prediction that sized them, and the excavation, forming and inspection cost of 25 separate pits is already comparable with a slab. A raft should be priced alongside the pad scheme before the design is fixed, and if the tolerable settlement cannot be relaxed the raft is likely to win.

Second, settlement governs, not strength. The margin against bearing failure is 4.8; the margin against exceeding 15 mm is about 1.02. Every decision that follows should therefore be aimed at stiffness and at the reliability of the settlement estimate, not at strength. The 15 mm limit is itself tight — 25 mm is the conventional allowance for an isolated footing on sand — and if the structural engineer can accept 25 mm the pads reduce to about 2.4 m and the raft question goes away.

Third, the groundwater condition is onerous and must be verified. A water table at the natural ground surface means excavation to 1.5 m requires dewatering or a cut-off, that the base of every excavation is at risk of boiling and of construction disturbance which will loosen the very sand the design relies on, and that the completed structure carries an uplift check on any below-grade slab. A blinding or working slab should be placed immediately on excavation. If the water table is in fact seasonal, the design case must be the highest credible level, because a rise from depth to surface reduces the effective stresses by about half and increases settlement.

Fourth, the ground investigation is thin for a 600 m² building. One SPT profile has been offered. The CFEM would expect enough coverage to demonstrate lateral consistency — as a minimum a second and third sounding, and preferably CPT profiles, which would also settle whether the constant $N_{60}=16$ below 6 m is real or an artefact of a single test position.

Fifth, the uniform load assumption is provisional. The 10 kPa per storey with equal sharing between 25 columns gives every column 1200 kN. Real frames deliver perhaps 1.3 times that to interior columns and half to corner columns, so pads must be re-sized to the structural engineer’s actual reactions. Sizing every pad for the same pressure, rather than the same load, is what keeps differential settlement small.

Sixth, this is a seismic country. A loose-to-medium saturated sand with $N_{60}$ of 10 to 16 and the water table at the surface is exactly the profile that requires a liquefaction screening under the National Building Code of Canada seismic provisions. That check, not bearing capacity, may end up controlling the foundation concept.

Assumptions made, with justification (Notes 1 and 7 on the cover page require these to be stated): the 10 kPa per storey is a total, unfactored service load including self-weight, and is shared equally by the 25 columns, because no column schedule is given; the footings are square, rigid, and founded in the same stratum; the SPT values are already corrected for energy and overburden, as the question states, so no further $C_N$ correction is applied; the sand is clean and cohesionless, $c=0$; a general shear failure mode is assumed, which is appropriate for medium-dense sand at this depth, though for the loosest zone a local-shear reduction of the strength parameters would be conservative; a factor of safety of 3 on net ultimate bearing capacity, which is the conventional value for a shallow foundation with routine investigation; and the water table is taken at the natural ground surface throughout, as the question directs.