Atterberg limits: the flow curve and the plasticity chart

A fine-grained soil does not have a liquid limit the way it has a density. It has a water content at which a groove of specified shape, cut in a cup of specified geometry, closes over a specified length after a specified number of drops. Change the apparatus and the number changes. That is not a criticism — it is the reason the test is useful. The limits are reproducible precisely because they are procedural, and a procedural definition is the only kind that survives being run by two laboratories on two continents.

What follows is one sample carried through the whole sequence: four cup trials, a fitted flow curve, the liquid limit read off it, and the classification that results. The coarse-grained half of the same problem — sieve sizes, uniformity, the logarithmic interpolation that trips people up — is worked through in soil classification from a grading curve.

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 coordinates are checked against the same arithmetic.

The three limitsPermalink to “The three limits”#

Atterberg’s limits mark the water contents at which a remoulded fine-grained soil passes between four states. Dry to wet:

Only two of them are used routinely, and they are used as a difference. The plasticity index

$$I_P = w_L - w_P$$

is the width of the water-content band over which the soil behaves plastically. A soil with a wide band deforms without cracking over a large range of moisture; a soil with a narrow one passes from stiff to slurry over a few percent. That range, not either limit alone, is what the classification systems are built on.

The flow curvePermalink to “The flow curve”#

A single cup trial is not a liquid limit. The blow count required to close the groove falls as water content rises, and no operator can mix a sample to land on exactly 25 blows. The test is therefore run at three or four water contents that bracket 25, and the liquid limit is interpolated.

The interpolation is linear in $\log_{10} N$, not in $N$:

$$w = a - I_f \log_{10} N$$

where $I_f$ is the flow index, the drop in water content per tenfold increase in blow count. Four trials on the sample:

Least squares on the pairs $(\log_{10} N,\ w)$ gives a slope of $-18.4661$ and an intercept of $73.4629$, so

$$I_f = 18.47, \qquad w = 73.4629 - 18.4661 \log_{10} N$$

and the liquid limit is the value at $N = 25$:

$$w_L = 73.4629 - 18.4661 \times \log_{10} 25 = 47.6\%$$

Note what the flow index is doing here. It is a slope, and it is also a sensitivity: a soil with a high flow index changes consistency rapidly for a small change in water content, which is worth knowing about a material that will sit under a fluctuating water table. It is computed on the way to the liquid limit and then, almost always, discarded.

The one-point method, and its costPermalink to “The one-point method, and its cost”#

Where a standard permits it, the liquid limit may be taken from a single trial whose blow count lies close enough to 25, corrected by

$$w_L = w_N \left(\frac{N}{25}\right)^{0.121}$$

Applying it to the third trial — $w_N = 48.9\%$ at $N = 22$ — gives 48.1%, against the 47.6% from the full fit. The difference is half a percentage point, which sounds negligible and is not always: it moves the plasticity index by the same amount, and a soil sitting within half a point of the A-line changes symbol on the strength of it.

The exponent is empirical. It encodes an assumed flow index, which is exactly the quantity a one-point test declines to measure. Use it when the alternative is no test; do not use it to classify a soil that is close to a boundary.

Plasticity index and the chartPermalink to “Plasticity index and the chart”#

With the plastic limit measured at 23.1% and 23.7% on two threads, $w_P = 23.4\%$, and

$$I_P = 47.6 - 23.4 = 24.2$$

Classification of fine-grained soils is then a position on the Casagrande plasticity chart: $I_P$ against $w_L$, divided by two straight lines.

The A-line separates clays from silts and organic soils:

$$I_P = 0.73\,(w_L - 20)$$

for $w_L$ above about 25.5, with a horizontal portion at $I_P = 4$ below it. The U-line is an empirical upper bound — no natural soil has been found reliably above it, so a point plotting above the U-line is a laboratory error before it is a discovery:

$$I_P = 0.9\,(w_L - 8)$$

for $w_L$ of 16 and above; below that the U-line is vertical at $w_L = 16$, running from the axis up to the $I_P$ of 7.2 at which the sloping portion begins. Nothing is classified to the left of it.

At $w_L = 47.6$ the two lines sit at

$$I_P^{\,A} = 0.73 \times 27.6 = 20.2, \qquad I_P^{\,U} = 0.9 \times 39.6 = 35.7$$

The sample’s plasticity index of 24.2 is above the A-line and below the U-line, and its liquid limit is below 50. That is a lean clay, CL.

Why the A-line sits where it doesPermalink to “Why the A-line sits where it does”#

The A-line is not derived from anything. Casagrande drew it through a body of test data in which soils of similar geological origin clustered, and its equation is a fit to that cloud. This matters when a soil plots close to it: the line summarises what had been tested by the late 1940s, not a physical boundary, and a point four tenths of a unit above it is not a clay in any sense a mineralogist would recognise.

What the position does carry is real. Above the line, plasticity comes mostly from clay minerals; below it, from particle shape and from organic content. Two soils with the same liquid limit on opposite sides of the line behave differently on drying, compact differently, and have different residual strengths. That is worth a symbol. It is not worth a decision made on the third significant figure.

The state the soil is actually inPermalink to “The state the soil is actually in”#

The limits describe a remoulded soil. They say nothing about the sample in the ground, which has a structure, a stress history and a natural water content that the test destroys before it begins. The bridge back is the liquidity index:

$$I_L = \frac{w_n - w_P}{I_P}$$

At a natural water content of 39.2%,

$$I_L = \frac{39.2 - 23.4}{24.2} = 0.65$$

A liquidity index near zero means the soil sits at its plastic limit and is stiff; near one, it is at its liquid limit and is soft; above one, it is holding more water than a remoulded sample could and is relying on structure to stand up — the condition in which a sensitive clay loses most of its strength when disturbed. At 0.65 this sample is soft to firm and unremarkable.

Activity completes the picture by normalising plasticity by how much clay is actually present:

$$A = \frac{I_P}{\text{fraction finer than 2 μm}} = \frac{24.2}{34} = 0.71$$

Below about 0.75 a soil is called inactive, which points at kaolinite or illite rather than at smectite. It is a cheap indication of mineralogy from two tests that were run anyway, and it is the number to reach for when a soil’s plasticity seems out of proportion to its clay content — a high activity is the early warning for swelling behaviour that no classification symbol will give you.

What the classification does not doPermalink to “What the classification does not do”#

A symbol is an index, not a design parameter. CL tells you what to expect and what to test for; it does not give a strength, a stiffness or a permeability, and correlations that claim to convert one into the other carry scatter that is routinely larger than the effect being predicted. The strength still has to be measured, and interpreted through the stress state it was measured in — which is what a Mohr circle is for.

The other thing the chart does not do is tell you the test was run properly. A point above the U-line is the clearest signal available that something went wrong, and it is worth checking every time, because it costs two multiplications and a comparison against a number the laboratory has already reported.