Notes on Particle Size Distribution, Hydrometer Testing, and Atterberg Limits

Sieve analysis, grain-size distribution, and grading concepts

Purpose: describe soils and understand materials for engineering decisions; focus on particle size distribution (PSD) and Atterberg limits as the main lab tests for classification and behavior.
Context from lecture: yesterday covered sieve analysis for coarse-grained materials; today extends to Atterberg limits and how to interpret PSD curves for soil typing.

Particle size distribution and key size values

PSD curve uses particle size on the x-axis and percentage passing on the y-axis.
D50 is not the mean; it is the median particle diameter: 50% of particles are larger and 50% are smaller.
Other percentiles can be read similarly, e.g., D30: the particle size at which 30% of the soil is passing that size.
How to read D30 on PSD curve (example method):
- Find 30% on the y-axis, draw a horizontal line to intersect the curve, drop down to the x-axis to read the corresponding particle size.
- In the example discussed, D30 read from the curve is approximately 0.6 mm.
Important caveat: lab data may be sparse; you fit a best-fit curve to define PSD and interpolate values, so measured data may not match exactly.

Reading and interpreting a PSD

The PSD helps distinguish coarse-grained vs fine-grained soils and informs grading.
Coarse-grained soils with fines content < 10% are analyzed using sieve analysis; very fine materials are analyzed with hydrometer test.
The 60 μm line (0.060 mm) is a common cutoff between coarse and fine fractions: everything to the left is coarse, to the right is fine. Some plots may omit units; always label axes.
In practice:
- Coarse fraction: determined by sieve analysis.
- Fine fraction: determined by hydrometer test and related methods.

Uniformity and curvature: Cu and Cc, and how to use them to classify well-graded versus poorly graded

Definitions:
- Uniformity coefficient: $Cu= rac{D{60}}{D_{10}}$
- Coefficient of curvature: $Cc= rac{D{30}^2}{D{10}D{60}}$
How to interpret (two cases depending on gravel vs sand composition):
- If the soil has more gravel content than sand (gravel-dominant):
- Well-graded if $Cu>4$ and $1c<3$ ; otherwise poorly graded.
- If the soil has more sand than gravel (sand-dominant):
- Well-graded if $Cu>6$ and $1c<3$ ; otherwise poorly graded.
Practical note: these criteria are numerical ways to express the broad qualitative idea of a well-graded curve (broad range of sizes, no gaps) vs poorly graded (gap-graded or uniform).
The D60, D10, D30 values are read from the PSD curve; units cancel in Cu and Cc to yield a unitless ratio.
Boundaries: the 60 μm cutoff is specifically used to separate gravels, sands, silts, and clays conceptually; the Cu/Cc rules are applied to classify coarse-grained soils and are used in practice for guidance on well- vs poorly graded behavior.

Applying the grading concepts to soil type identification

For coarse-grained soils (fines content < 10%), use Cu and Cc with the appropriate thresholds to decide well-graded vs poorly graded.
For soils with more sands than gravels, the threshold for Cu is higher (Cu>6) while Cc remains 1–3 for a well-graded condition.
In the lab, you will describe soils as well-graded or poorly graded (for sand and gravel components) based on these criteria.

Practical limits of sieve and why hydrometer tests exist

Sieves are expensive; a typical 60 μm sieve costs a few hundred dollars, which motivates using hydrometer tests for finer fractions.
Hydrometer test extends the PSD to fines below 60 μm; it uses sedimentation principles and Stokes’ law for spheres in a fluid.
The two-stage approach: sieve analysis for the coarse fraction; hydrometer for the fines, and another test to distinguish silt vs clay in the fines.
Expect to see a combined PSD curve: the sieve-derived portion (coarse) and the hydrometer-derived portion (fine). The purple line (hydrometer) complements the blue sieve line.

Hydrometer test and fine-grained soils

Concept: sedimentation velocity depends on particle diameter; larger particles settle faster than smaller ones.
Simplifying assumption: treat all soil grains as spheres (a reasonable engineering simplification).
Settling velocity intuition: larger particles settle quickly; very fine particles settle slowly; the distribution of settled material with time yields the PSD for fines.
Key relation (sedimentation velocity): velocity is proportional to the square of the particle diameter:
- Heuristic: larger particles settle out of suspension faster than finer particles.
The hydrometer instrument measures the density of the suspension as particles settle; density trends toward water density as solids leave suspension.
Typical soil specific gravity: $S_g ext{ (specific gravity of solids)} \approx 2.65$ (varies with mineralogy).
As heavier solids settle, the suspension density decreases toward the density of water, and the hydrometer reading drops accordingly.
Conceptual setup:
- A cylinder with soil-water suspension is agitated; over time, particles settle to different depths.
- At a given time, a sample is taken from a fixed height in the column to determine the mass of soil in that sample and water mass, yielding a snapshot of the PSD at that time.
- Repeated sampling over time builds the PSD curve, particularly for fine fractions.
Hydrometer readings and sedimentation: you measure the density of the suspension at various times; the change in density over time relates to the fraction of fine particles in suspension that have not yet settled.
Practical notes:
- Using a hydrometer avoids the fiddly task of repeatedly pipetting samples and disturbing flows; it provides a robust, practical approach to fines analysis.
- The method is especially useful for very fine fractions (below 60 μm).

Reading and interpreting the hydrometer-derived PSD

The hydrometer curve is plotted by turning time into a distribution of particle sizes. Heuristics link settling time to particle size via Stokes’ law.
The classic cutoff (60 μm) separates the sieve and hydrometer analyses; the hydrometer handles the finer end of the spectrum.
The lab workflow: you perform sieve analysis for the coarse portion and hydrometer analysis for the fines; the combination yields the full PSD.
Practical lab points:
- You will not perform full hydrometer tests in the current lab due to time; conceptually, you should understand how the hydrometer informs the fines distribution.
- It’s common to see a schematic PSD with a blue line (sieve data) and a purple line (hydrometer data) joined to form a complete curve.
- If a lab lacks perfect 60 μm sieves, a nearby size (e.g., 75 μm) can be used with leniency in calculations.

Schematic workflow for PSD construction in lab practice

Step 1: conduct sieve analysis with a stack of sieves; record masses passing each sieve.
Step 2: compute the percent passing to generate discrete PSD points (coarse fraction).
Step 3: perform hydrometer testing on the fraction remaining below the finest sieve (or the portion passing the 60 μm sieve) to determine the fines distribution (silt vs clay behavior discussed separately).
Step 4: plot PSD with the sieve data (blue) and the hydrometer data (purple) and join them to form a full curve.
Step 5: interpret the PSD to classify the soil type and to decide if the soil is well-graded or poorly graded, and if it contains significant fines.
Step 6: classify the fines as silt or clay using additional tests (discussed next).

Atterberg limits and plasticity concepts

Focus: For fine-grained soils (silt and clay), Atterberg limits characterize how moisture content controls physical behavior (plasticity, workability, and strength).
Important note on definitions:
- Water content (often called moisture content): $w = rac{Mw}{Ms}$ where $Mw$ is the mass of water and $Ms$ is the mass of solids; in practice, this is measured by weighing a sample before and after oven-drying.
- In practice, the terms water content and moisture content are interchangeable in this course.
Three soil phases to keep in mind: solids, voids, and void filling (air or water). Moisture content relates to how much liquid is in the voids.
The three key Atterberg limits (and the range between LL and PL) define the plasticity characteristics of fine-grained soils:
- Liquid limit (LL): the water content at which soil begins to behave as a liquid (flows) when subjected to a standard energy input.
- Plastic limit (PL): the water content at which soil begins to behave plastically and can be remolded with fingers.
- Plasticity index (PI): $PI = LL - PL.$ A larger PI indicates a more plastic soil (more clay content).
The liquidity index (LI) relates field moisture to the laboratory-defined limits:
- $LI = rac{w - PL}{LL - PL}$ where $w$ is the in-situ moisture content.
- Interpretations:
- $LI > 1$ : the soil is very moist in-situ, often indicating a sensitive soil with low strength.
- $LI ightarrow 1$ : clayey behavior near the liquid limit.
- $LI ightarrow 0$ : soil near the plastic limit, relatively stable and plastic.
Behavior with moisture content: as water increases, soil moves to the right on the Atterberg limits chart, from solid-like to plastic-like to liquid-like behavior.

Tests to determine LL and PL

Liquid limit tests (two common methods):
- Casagrande test (classically used in the US): dynamic and widely used; uses a groove cut in a moist soil sample in a brass cup; standard procedure:
- Groove width: 2 mm; the groove is closed by dropping the cup from a fixed height; the number of blows required to close the groove is counted.
- The moisture content at the blow count of 25 (the standard) is the liquid limit: this is called LL.
- Typical phrasing: prepare the specimen, perform the test, and record the moisture content when the groove closes after 25 blows.
- Variability: not every specimen yields exactly 25 blows; often you perform multiple trials and fit a line to data to interpolate LL at 25 blows.
- Liquid limit cone penetrator (static approach): cone method with a cone of 80 g, 30° apex angle, 30 mm diameter; the cone penetrates into the soil 20 mm in 5 seconds; the corresponding moisture content is read from a calibration curve; this is considered slightly more stable due to its static nature.
Plastic limit tests:
- Plastic limit test (often called the thread or worm method): roll a thread of soil between the fingers until it cracks at a diameter of 3 mm; the moisture content at the point of cracking is the plastic limit PL.
- If the thread does not crack at 3 mm, the soil is rolled further or dried until a crack occurs; the measured water content at cracking is PL.
- This test measures the brittle-to-plastic transition in the soil as it dries.
Practical notes on LL and PL:
- In practice, you perform multiple LL tests (often 3–4) at different moisture contents to obtain a reliable LL by plotting and interpolating at the 25-blow point.
- Similarly, PL is determined with repeatability and careful handling to ensure consistent results.
- Organic soils and volcanic clays behave differently: liquid limits can be >100% (water content exceeds the mass of solids); repeated wetting-drying cycles can alter LL, PL, and PI; volcanic soils are particularly sensitive to wetting/drying cycles, making standardized preparation essential.
- New Zealand vs United States distinctions: NZ emphasizes plasticity within a given fine fraction; US uses a more particle-size-based approach with the A-line and U-line to classify clays vs silts.

Interpreting Atterberg limits and soil type from LL and PL

Stability and classification using the plasticity index (PI):
- Low PI (0–3): typically non-plastic soils (often silts with low plasticity or some sands in borderline cases).
- Moderate PI (3–10, approximate): low-plasticity clays or silts.
- High PI (>30): highly plastic soils (strong clays).
- General rule: the larger the PI, the more clay-sized material and the more plastic the soil.
Liquidity index interpretation and field strength:
- LI > 1 indicates a very moist, weak, or potentially sensitive soil in-situ.
- LI near 0 indicates a soil near the plastic limit, with stable mechanical behavior under typical conditions.
Graphical classification tools:
- A-line and U-line chart (United States approach): a plot of PI vs LL with an A-line that separates clays from silts; soils below the A-line tend to be silts, soils above tend to be clays; the U-line marks a boundary beyond which measurements may be in error or unusual.
- NZ approach emphasizes behavior and may differ in how it uses particle size thresholds (e.g., using 60 μm as a cutoff for defining clay in some NZ contexts).
Field-to-lab linkage:
- Liquidity index relates in-situ moisture to LL and PL to assess strength and potential consolidation behavior in the field.
- For field and laboratory comparisons, expect variability: soils are inherently variable, so report ranges of LL, PL, and PI for large deposits rather than a single value.
Special soil types and caveats:
- Organic soils: high LL, high variability; LL can exceed 100%; cycling wetting/drying can alter LL, PL, and PI; require rigorous preparation and testing.
- Volcanic clays: particularly sensitive to wetting/drying cycles; require careful prep and testing to obtain representative limits.

Classification and lab workflow in geotechnical practice

Overall workflow:
- Step 1: sieve analysis to characterize the coarse fraction (gravel and sand) and obtain PSD points for the coarse end.
- Step 2: hydrometer test to characterize the fine fraction (< 60 μm) and obtain PSD points for the fines end.
- Step 3: combine data from sieve and hydrometer to construct a complete PSD curve.
- Step 4: determine whether the soil is predominantly fine- or coarse-grained and whether it is well-graded or poorly graded based on Cu, Cc and the gravel/sand distribution.
- Step 5: perform Atterberg limit tests (LL, PL) to determine PI, and use LI to assess in-situ moisture effects.
- Step 6: classify soils using USCS (Unified Soil Classification System) or NZ approaches as appropriate, using the A-line and U-line for fine-grained soils and PSD-based criteria for coarse-grained soils.
Practical notes for the lab:
- You will perform sieve analysis next week; expect to record the mass passing each sieve and compute percent passing.
- For the hydrometer test, you will conceptually understand the procedure and the PSD interpretation; in practice, the hydrometer will be used to extract the fines distribution.
- Always label axes on PSD plots with proper units; remember the 60 μm boundary and how to interpret the two portions of the curve.
- If you encounter lab equipment limitations (e.g., lack of a precise 60 μm sieve), discuss with TAs; using a nearby nominal size and documenting deviations is common in teaching labs.
- Record variability and report a range for LL, PL, PI for soil deposits that may vary across boreholes or test sites.
Real-world relevance and implications:
- PSD and Atterberg limits influence drainage and permeability (water movement through soils), shear strength, and overall stability of soil structures.
- Grading quality (well-graded vs poorly graded) affects how soils drain and compact; coarse-grained well-graded soils tend to have higher shear strength and favorable drainage, while poorly graded soils may be more prone to settlement and strength loss under varying moisture contents.
- Fine-grained soils with high PI are more plastic and may require careful design of fill and foundations to account for movement and strength changes with moisture.

Quick recap of key formulas and critical values to memorize

Percentiles and PSD readouts:
- $D_{50}$ : median particle size (50% passing).
- $D{10}, D{30}, D_{60}$ : particle sizes corresponding to 10%, 30%, 60% passing, respectively.
- Example: $D_{30} ext{ read from PSD curve in the example} \approx 0.60 ext{ mm}.$
Uniformity and curvature:
- $Cu= rac{D{60}}{D_{10}}$
- $Cc= rac{D{30}^2}{D{10}D{60}}$
Well-graded vs poorly graded criteria (coarse-grained):
- Gravel-dominant: well-graded if $Cu>4$ and $1c<3$ ; otherwise poorly graded.
- Sand-dominant: well-graded if $Cu>6$ and $1c<3$ ; otherwise poorly graded.
60 μm boundary: $60 ext{ μm} = 0.060 ext{ mm}$ .
Atterberg limits:
- Liquid limit LL: moisture content at which a groove closes after 25 blows in the Casagrande test, or via LL-cone method.
- Plastic limit PL: moisture content at which soil threads crack at 3 mm diameter.
- Plasticity index: $PI = LL - PL$ .
- Liquidity index: $LI = rac{w - PL}{LL - PL}$ , where $w$ is the in-situ water content.
Hydrometer hydrodynamics (concept): settling velocity ~ diameter squared; in full expression (Stokes’ Law for spheres):
- $v = rac{( hos - hof) g d^2}{18 u}$ or equivalent form depending on viscosity; qualitative relation: larger particles settle faster.

If you want, I can tailor these notes to focus more on a specific section (e.g., more detailed step-by-step LL testing procedure, or more emphasis on USCS vs NZ classifications) or add a compact one-page cheat sheet with the most-used formulas and decision criteria. Also let me know if you’d prefer fewer or more examples in the PSD reading section.