Question 7 of 9: Corrected SPT Profile in a Sand Deposit

Question 7: Corrected SPT Profile in a Sand Deposit (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. (i) the overburden- and energy-corrected blow count $(N_1)_{60}$ at each depth and the single average value to be carried into design; and (ii)–(iv) the engineering interpretation of that profile.

Approach. Build the vertical effective stress profile first, then apply the two corrections in the order energy →  overburden: $N_{60} = N_F\eta_H\eta_B\eta_S\eta_R/60$ followed by $(N_1)_{60} = C_N N_{60}$ with Skempton's $C_N = 2/(1+\sigma'_v/p_a)$. The corrected profile is then read for relative density, friction angle and settlement, and finally weighed against a $\phi'$-based bearing-capacity calculation.

Part (i) — the corrected blow count

1. Part (i) — build the effective stress profile. Above the water table the sand is dry, below it the buoyant unit weight applies: $$\gamma' = \gamma_{sat} - \gamma_w = 20.1 - 9.81 = 10.29\ \text{kN/m}^3 .$$ Hence $\sigma'_v = \gamma_d z$ for $z \le 3$ m and $\sigma'_v = 17.5(3) + 10.29(z-3)$ below, giving 35.0, 62.8, 83.4, 104.0, 124.5 and 145.1 kPa at the six test depths. The effective stress does not double from 6 m to 12 m even though the depth does, because buoyancy removes almost half the weight of every metre below 3 m.
2. Apply the energy correction. The general form is $$N_{60} = \frac{N_F\,\eta_H\,\eta_B\,\eta_S\,\eta_R}{60},$$ where $\eta_H$ is the hammer efficiency in per cent, $\eta_B$ the borehole diameter factor, $\eta_S$ the sampler factor and $\eta_R$ the rod-length factor. Assuming a safety hammer with a rope-and-cathead system delivering $\eta_H = 60$, a standard 100 mm borehole ($\eta_B = 1.0$) and a standard sampler without liners ($\eta_S = 1.0$) — the normal Canadian rig configuration — the expression collapses to $N_{60} = N_F\,\eta_R$. Taking Das's rod-length factors (0.75 for 0–4 m, 0.85 for 4–6 m, 0.95 for 6–10 m and 1.00 beyond 10 m), $N_{60}$ becomes 7.5, 10.2, 13.3, 15.2, 16.2 and 18.0. The rod correction bites hardest at shallow depth, where the short rod string reflects much of the hammer energy back up.
3. Apply Skempton's overburden correction. For a normally consolidated fine sand, $$C_N = \frac{2}{1 + \sigma'_v/p_a}, \qquad p_a = 100\ \text{kPa},$$ which returns $C_N = 1.481$ at 2 m, falling to 0.816 at 12 m. The factor is greater than one where the confining stress is below one atmosphere and less than one where it is above, which is exactly the bias it is meant to remove.
4. Tabulate and average. Multiplying, $(N_1)_{60} = C_N N_{60}$, and averaging the six values, $$\bar{(N_1)}_{60} = \frac{11.11+12.53+14.51+14.91+14.39+14.69}{6} = 13.69 ,$$ so that $$\boxed{\,(N_1)_{60,\,ave} \approx 14\,}$$ is the value recommended for design. The individual results and their inputs are set out below.

Part (ii) — what the corrected profile tells us

The field record climbs from 10 to 18 blows, an 80 per cent increase, and an inexperienced reader would conclude that the sand becomes markedly denser with depth. The corrected record does not: from 4 m downward $(N_1)_{60}$ sits between 12.5 and 14.9, a spread of about 3.8 blows over the whole profile, and the trend is essentially flat. The apparent improvement with depth is an artefact of increasing confining stress, not a real change in density. The deposit is a single, uniform, medium dense sand. That is a materially different design picture: it means no stratum boundary need be modelled, and it means the shallow sand is no weaker in relative-density terms than the deep sand.

Quantitatively, with $(N_1)_{60} = 14$ the deposit classifies as medium dense on the Terzaghi–Peck scale. Skempton's relation $D_r = \sqrt{(N_1)_{60}/60}$ gives $D_r = 48$ per cent, and Wolff's correlation $$\phi' = 27.1 + 0.3(N_1)_{60} - 0.00054\left[(N_1)_{60}\right]^2 = 27.1 + 4.11 - 0.11 = 31.1^{\circ}$$ puts the effective friction angle at about $31^{\circ}$, which Peck–Hanson–Thornburn's chart confirms for the same blow count. For shear strength behaviour this matters: at $D_r \approx 48$ per cent the sand is close to its critical state, so it will neither dilate strongly nor collapse, and a peak–residual distinction is unimportant. The deposit is, however, loose enough to be a liquefaction candidate under seismic loading in a coastal British Columbia setting, and $(N_1)_{60} = 14$ sits below the commonly used screening threshold of 30, so a cyclic-resistance evaluation would be required for any significant structure.

For shallow foundation design the profile supports two things. First, a bearing-capacity calculation using $\phi' = 31^{\circ}$. Second, and more important, a settlement calculation: for a footing of width $B = 2$ m at 2 m depth the average $N_{60}$ over the $2B$ influence zone is 10.3, and Meyerhof's direct rule for 25 mm of settlement, $$q_{all} = 11.98\,N_{60}\left[\frac{3.28B+1}{3.28B}\right]^2 = 11.98(10.33)\left[\frac{7.56}{6.56}\right]^2 = 164\ \text{kPa},$$ gives an allowable pressure of about 165 kPa. In a medium dense sand of this kind settlement almost always governs before bearing capacity does, so this is the number that sizes the footing.

Part (iii) — why the corrections are necessary

The SPT is not a measurement of a soil property; it is a measurement of the response of one particular apparatus, operated in one particular way, at one particular depth. Two corrections are needed to turn it into something transferable.

Energy. The blow count is inversely proportional to the energy actually delivered to the sampler, and delivered energy varies from about 45 per cent of theoretical for a donut hammer with two rope turns to about 80 per cent for an automatic trip hammer. The same sand tested by two crews on the same day can return blow counts differing by nearly a factor of two. Because every published correlation — for $D_r$, $\phi'$, $E_s$, liquefaction resistance — was calibrated against a nominal 60 per cent energy ratio, using an uncorrected $N_F$ in those correlations imports whatever the driller's rig happened to deliver directly into the design. Rod length, borehole diameter and the presence of sampler liners each add a further 10 to 25 per cent.

Overburden. Penetration resistance rises with confining stress even at constant density. Uncorrected, a uniform deposit looks as though it improves with depth, and two sites of identical density but different water tables appear to be different soils. Skempton's $C_N$ normalises every reading to a reference effective stress of 100 kPa so that blow counts from different depths, different sites and different groundwater conditions can be compared, and so that a design value can legitimately be averaged over a profile — which is precisely what part (i) does. From an engineering-practice point of view the corrections are what make the SPT auditable: they let a reviewer reproduce the design from the borehole log, and they let a Canadian consultant use a correlation published from American or Japanese data.

Part (iv) — Meyerhof's equation, or the corrected SPT?

Use both, but let the corrected SPT govern the design. The two routes answer different questions, and in a medium dense sand they do not compete on equal terms.

Meyerhof's general bearing capacity equation $q_u = qN_qF_{qs}F_{qd}F_{qi} + \tfrac12\gamma BN_{\gamma}F_{\gamma s}F_{\gamma d}F_{\gamma i}$ answers the ultimate limit state: it tells you the pressure at which the sand fails in shear. Its difficulty here is that its only real input, $\phi'$, is itself obtained from the corrected SPT through a correlation carrying $\pm 2$ to $3^{\circ}$ of scatter — and because $N_{\gamma}$ and $N_q$ grow roughly exponentially with $\phi'$, a $3^{\circ}$ error changes $q_u$ by 30 to 40 per cent. The equation is not more rigorous than the SPT data behind it; it is the same data pushed through a sensitive amplifier. Worse, it says nothing at all about deformation.

The corrected SPT used through a settlement rule — Meyerhof's 25 mm expression above, Burland and Burbidge's method, or Schmertmann with $E_s = 500(N_{60}+15)$ kPa — answers the serviceability limit state, and in sand that is what controls. For the 2 m footing considered above, $\phi' = 31^{\circ}$ through the bearing-capacity equation would permit several hundred kilopascals at $FS = 3$, whereas the 25 mm criterion permits only 165 kPa. Settlement governs by a wide margin, and it will do so for any footing of ordinary size on a medium dense sand.

The practical recommendation, and the one consistent with the Canadian Foundation Engineering Manual's limit-states framework, is therefore to size the footing on the corrected SPT settlement criterion, then verify the ultimate limit state with Meyerhof's equation using $\phi' = 31^{\circ}$ and a global factor of safety of 3 (or the CFEM resistance factor of 0.5), and adopt whichever is smaller. In this deposit that will be the settlement value.