CPT interpretation: from qc and fs to Ic and su

A cone penetration test measures three things: the force on the cone tip, the force on the friction sleeve, and a water pressure behind the shoulder. Everything else — soil type, undrained strength, relative density, stiffness — is inferred, and each inference is a step in a chain where the assumptions accumulate.

The chain is short enough to write out in full, which is what this article does, for one reading. The point of doing it in full is that the result turns out to depend on a property of the cone.

Every number in this article is recomputed from its inputs by a script that runs on each build of this site, and the chart’s geometry is checked against the same arithmetic.

The readingPermalink to “The reading”#

At 12.0 m in a sounding through a soft deposit:

With an average unit weight of 18.5 kN/m³ and the water table at 1.5 m, the in-situ stresses at that depth are

$$\sigma_v = 12.0 \times 18.5 = 222.0\ \text{kPa}, \qquad u_0 = 10.5 \times 9.81 = 103.005\ \text{kPa}$$ $$\sigma'_v = 222.0 - 103.005 = 118.995\ \text{kPa}$$

Which is the same subtraction as always, and it is worth noticing that the CPT chain needs an assumed unit weight profile before it can produce anything — the test that is supposed to characterise the soil requires a characterisation of the soil to be interpreted.

Step one: correct for the cone’s own geometryPermalink to “Step one: correct for the cone’s own geometry”#

The load cell behind the cone tip does not see the whole tip area. The shaft passes through a seal of smaller diameter, so the water pressure acting on the back of the cone carries part of the load that the tip is credited with. The net area ratio $a$ is the fraction of the tip area that is structurally continuous, and the corrected resistance is

$$q_t = q_c + u_2\,(1 - a) = 1420 + 780 \times 0.20 = 1576.0\ \text{kPa}$$

That is a correction of 156 kPa, 11% of the raw reading, applied because of how the cone was built. In free-draining sand $u_2$ is small and the correction is negligible. In soft clay, where $u_2$ can exceed $q_c$, it is not optional, and this is where cone data is most often used.

$a$ is a calibration property of the individual cone. It should appear on the calibration certificate. It routinely does not appear in the data file.

Step two: normalisePermalink to “Step two: normalise”#

Raw resistance grows with depth because the soil is confined more strongly, not because it is stronger in any intrinsic sense. Robertson’s normalisation removes the overburden:

$$Q_t = \frac{q_t - \sigma_v}{\sigma'_v} = \frac{1354.0}{118.995} = 11.38$$ $$F_r = \frac{f_s}{q_t - \sigma_v}\times 100 = \frac{42}{1354.0}\times 100 = 3.10\%$$

$Q_t$ is dimensionless resistance; $F_r$ is friction expressed as a percentage of net resistance. Together they place the reading on the soil behaviour type chart.

Step three: the indexPermalink to “Step three: the index”#

The chart’s zones are, over most of their extent, concentric circular arcs about a single point, and the soil behaviour type index is just the radius:

$$I_c = \sqrt{\left(3.47 - \log_{10} Q_t\right)^2 + \left(\log_{10} F_r + 1.22\right)^2}$$ $$I_c = \sqrt{(3.47 - 1.0561)^2 + (0.4916 + 1.22)^2} = 2.959$$

The boundaries used with it are $I_c = 2.60$ between sand mixtures and silt mixtures, $I_c = 2.95$ between silt mixtures and clays, and $I_c = 3.60$ between clays and organic soils. At 2.959 this reading falls in the clays, by a margin of 0.009 in the index.

One point of definition, because two versions are in circulation. The index above is taken on $Q_t$, which is the normalisation with a stress exponent of one. The later formulation normalises instead by $Q_{tn} = \left[(q_t - \sigma_v)/p_a\right](p_a/\sigma'_v)^{n}$ with the exponent found iteratively from $n = 0.381 I_c + 0.05\,\sigma'_v/p_a - 0.15$, capped at one. Run that iteration here and it returns 1.0369, 1.0094 and 1.0652 for the three cases below — all above the cap, so $n = 1$, so $Q_{tn} = Q_t$ and the two definitions give the same number. The exponent falls towards 0.5 in sands, where they would not.

What the classification actually depends onPermalink to “What the classification actually depends on”#

Now change one thing. Keep every measured signal identical and suppose the cone had a net area ratio of 0.58 rather than 0.80 — both are values real cones are supplied with. Then

$$q_t = 1420 + 780 \times 0.42 = 1747.6\ \text{kPa}$$ $$Q_t = 12.82, \qquad F_r = 2.75\%, \qquad I_c = 2.887$$

The index has crossed 2.95 in the other direction. The same ground, sounded on the same day with the same three signals, is now silt mixtures rather than clays.

And if the correction is skipped altogether — $q_c$ used where $q_t$ belongs, which is what happens whenever $u_2$ is not recorded — the index goes to 3.033, further into the clays.

None of this is an argument against the CPT, which remains the most information-dense ground investigation tool available per metre of borehole. It is an argument against treating a soil behaviour type as a measurement. It is a classification of the response of the ground to a cone, computed through a chain that includes at least one number about the equipment, and near a zone boundary it is not resolvable at all.

Step four: strength, and the factor nobody measuresPermalink to “Step four: strength, and the factor nobody measures”#

The undrained shear strength follows from the net resistance and a cone factor:

$$s_u = \frac{q_t - \sigma_v}{N_{kt}}$$

$N_{kt}$ is not measured on the site. It is taken from a correlation, and published guidance puts it between about 10 and 20, with 14 a common default. Three values from inside that range:

A factor of 1.67 between the ends of those three, and 2.0 across the full published range, either end of which a competent engineer could defend. Against that, the 11.1 kPa the area correction contributes at $N_{kt} = 14$ — 13.02% of the uncorrected net resistance — looks small, and it is. It is also the part that is deterministic: it is known exactly once the cone’s calibration certificate is in hand, whereas $N_{kt}$ is genuinely uncertain until a laboratory strength is available to back-figure it against.

That distinction is the useful one. Uncertainty that can be removed by bookkeeping should be removed by bookkeeping, so that the uncertainty that remains is visible and can be argued about on its merits.

Why this belongs in a tool rather than a spreadsheetPermalink to “Why this belongs in a tool rather than a spreadsheet”#

The chain above has four steps, seven inputs, and two places where a value from the equipment certificate enters. Done by hand it is twenty minutes per sounding and is usually done once, at a few selected depths, by someone reading values off a plot.

Done as a program it runs at every measured depth — a CPT logs every 20 mm — and, more importantly, it records which $a$, which $N_{kt}$ and which unit weight profile produced each number. A year later, when the strength profile is questioned, that record is the difference between re-deriving the answer and re-doing the investigation. It is the same argument as for recomputing the numbers in an article at build time: a derivation that cannot be replayed is not a derivation, it is a recollection.