Consolidation Test Curve (e–log σ′): Casagrande Preconsolidation Pressure, Free

Casagrande’s construction, step by step

Casagrande (1936) gave the construction still used in practice. Find the point of maximum curvature on the curve. Through it draw a horizontal line and the tangent to the curve, and bisect the angle between them. Extend the straight virgin compression line back until it meets the bisector. The stress at that intersection is σ′p.

This tool does exactly that, on the figure you see. The virgin line is a least-squares fit through the last few first-loading readings (three by default, adjustable). The curve between readings is a smooth spline, and the point of maximum curvature is searched for along it, between the first reading and the start of the virgin line. Unloading and reloading readings are kept off that curve — they are drawn hollow — so an unload–reload loop part-way through the test does not distort the bend. With the usual doubling of load between readings, the sharpest bend of that spline usually falls on a reading rather than between two, which is one more reason the band below matters.

Why the pick is shown, and why you can move it

Curvature is not a property of the data alone. Stretch the void-ratio axis and the sharpest-looking bend of the same curve moves, and with it the tangent, the bisector and σ′p. A test sheet plotted by two people at two scales can give two answers by the same construction. That is a property of the method, not of any one implementation, and it is why a σ′p quoted without its figure is hard to check.

So this page measures curvature in one stated frame — the readings’ own axes, stress snapped out to whole decades and void ratio to round ticks, filling the plot at the aspect ratio it is drawn — and draws the point it found. It also reports a band: σ′p from every point where the curvature is within 10 % of its maximum. A narrow band means the bend is sharp and the answer is robust; a wide one means the judgement matters and you should say so. If your own reading of the curve puts the bend somewhere else, type that stress into the pick option: the construction is redrawn from your point, the automatic one stays on the figure for comparison, and the results table records that the pick was yours.

On a smoothly rounded curve Casagrande’s σ′p sits somewhat above the corner where the recompression and virgin lines would cross if you extended both — the geometry of the bisector puts it there. Other constructions exist, such as Pacheco Silva’s and energy-based methods; they give different numbers from the same data and are not implemented here. Quote the method with the value.

Sample disturbance and the Schmertmann field curve

Sampling, transport and trimming all disturb a clay, and disturbance does one consistent thing to the laboratory curve: it rounds off the bend and flattens the virgin line. Read straight off a disturbed specimen, σ′p comes out low and Cc comes out low — both on the unsafe side for a settlement estimate.

Schmertmann (1955) observed that the laboratory virgin lines of increasingly disturbed specimens of one clay converge near a void ratio of about 0.42 e₀, and used that to reconstruct the field curve. For a normally consolidated clay it is a straight line from the in-situ state (σ′v0, e₀) to the laboratory virgin line at 0.42 e₀. For an overconsolidated clay it runs from (σ′v0, e₀) parallel to the unload–reload loop up to σ′p, then straight to the same 0.42 e₀ point. The slope of that last leg is the field compression index, normally steeper than the laboratory Cc. Give a sample its σ′v0 (and its e₀) and the tool draws the field curve and tabulates the field Cc beside the laboratory one.

Who uses one

Civil and geotechnical engineering students writing up the consolidation lab, geotechnical laboratories reporting oedometer tests, and engineers estimating the settlement of foundations, embankments and fills on soft clay. Anyone checking a reported σ′p or OCR against the curve it was read from.

Common questions

What data do I need?

One row per load increment, in the order the test ran: the vertical effective stress and the void ratio at the end of that increment (end of primary consolidation, or the end-of-increment reading your lab reports). Keep the unloading and reloading steps in their place in the sequence — a stress lower than the one before is read as unloading. A `sample` column lets several tests share the table. If your sheet reports strain rather than void ratio, put it in the strain column (per cent) and give the sample an e₀; void ratio is then e = e₀ − ε(1 + e₀).

Where do e₀ and σ′v0 go?

On any one row of the sample — they are per-sample values, so the rest of that column can stay blank. e₀ is the initial void ratio of the specimen; without it the tool uses a zero-stress seating reading if there is one, otherwise the first reading, and says which in the tooltip. σ′v0 is the present vertical effective stress at the sample depth. It is needed for the OCR and the Schmertmann field curve, and nothing else.

Why did σ′p change when I changed the number of virgin-line readings?

Because the bisector is run into the virgin line, and the virgin line is fitted through those readings. If one of them is still on the bend, the line comes out too flat and σ′p moves. Use readings that are clearly on the straight part — the figure draws the line, so you can see whether it fits.

How is Cr worked out?

From the first unloading in the test. If the test reloads after it, Cr is the mean slope of the unload–reload loop over its stress range, which is the slope Schmertmann’s construction uses. If the only unloading is the final rebound, Cr is the slope of that branch alone, and the table says so. With no unloading readings there is no Cr, and an overconsolidated field curve cannot be drawn.

Does the stress unit matter?

Only for the labels. Every result here is a ratio of stresses or a slope per log cycle, so σ′p comes out in the unit you entered. The unit option changes the axis title and the table headers; it never converts your numbers.

Has this been checked against a published worked example?

Not yet, and there is deliberately no entry for it on the validation page. A fair comparison needs a published example with its readings and its σ′p both stated, and a construction that depends on a hand-picked point is hard to compare to a figure read by eye. What the test suite does instead is internal: it plants a known σ′p in synthetic data and checks the construction recovers it when the bend is sharp; it checks the automatic pick lands at the geometric vertex of a curve whose vertex is known exactly; and it checks Cc, Cr and the Schmertmann points against the lines they were built from.

Where does the sample data come from?

It is synthetic, and the sample names start with SYN- to say so. Both records come from a stated model whose σ′p is an input, with ordinary clay parameters, a doubling load schedule, three decimals of void ratio and a little seeded scatter. A table typed to look like a real test would be presenting an invention as a measurement.

Is this free? Does anything leave my browser?

Free, with no account and nothing gated, and nothing is uploaded — the analysis runs in this tab. Export the figure as SVG or PNG; free exports carry a small watermark, as they do everywhere on this site.