Grain Size Distribution Curve

Theory: Grain Size

How ASTM D6913 and D7928 determine the grain-size distribution by sieving and hydrometer analysis, and how to read the grain-size curve.

Grain-size analysis

Grain-size analysis determines the proportion, by mass, of particles of each size in a soil. The result is the grain-size curve: the percent finer than each diameter against diameter, on a log scale, since sizes range from centimeters to micrometers.

ASTM D6913 covers sieving down to the No. 200 (75 µm) sieve, and D7928 the finer particles, by sedimentation with a hydrometer. Here the specimen is split on the No. 10 (2.00 mm) sieve, as D6913's composite sieving allows: the coarse portion is washed and sieved whole, and a subspecimen of the portion passing goes to the hydrometer test, if there are fines to characterize, before being washed on the No. 200 sieve and sieved.

Sieving

The total dry mass combines the coarse portion, already dry, with the portion passing No. 10, corrected for its hygroscopic water content:

M_d = \frac{(M_t - M_c) \cdot 100}{100 + w} + M_c

Mt: total moist mass · Mc: dry mass retained on the No. 10 sieve · w: hygroscopic water content (%)

\mathrm{PF} = \frac{M_d - \sum M_r}{M_d} \cdot 100

Coarse sieving · PF: percent finer · ΣMr: cumulative mass retained down to each sieve

\mathrm{PF} = N \left(1 - \frac{\sum M_r}{M_s}\right)

Fine sieving · Ms: dry mass of the subspecimen, from its moist mass and w · N: percent finer than the No. 10 sieve

Portion	Sieves
Coarse	3 in. · 2 in. · 1½ in. · 1 in. · ¾ in. · ⅜ in. · No. 4 · No. 10
Fine	No. 20 · No. 40 · No. 60 · No. 100 · No. 140 · No. 200

Hydrometer analysis

Particles finer than the No. 200 sieve are sized by sedimentation. The subspecimen is dispersed with sodium hexametaphosphate, made up to 1000 mL with water and left to settle in a cylinder. By Stokes' law, each particle falls at a constant velocity proportional to the square of its diameter: at time t and effective depth Hm, only particles finer than this remain in suspension

D = \sqrt{\frac{18\, \eta\, H_m}{(\rho_s - \rho_w)\, g\, t}}

D: particle diameter · η: viscosity of water · ρs = Gs · ρw: density of the solids · ρw: density of water at the test temperature · Hm: effective depth · t: elapsed time

The 152H hydrometer reads grams of soil per litre, for Gs = 2.65 in water at 20 °C. Its reading R, corrected for the meniscus Cm, gives the effective depth from the calibration (nominally Hm = 16.3 − 0.164 R cm). The reading minus that in a control cylinder of dispersant solution at the same temperature, Rd, gives the mass still in suspension, which already corrects for the dispersant and temperature:

\mathrm{PF} = N\, \frac{G_s}{G_s - 1} \cdot \frac{1.65}{2.65} \cdot \frac{R - R_d}{M_s}

R: reading in the suspension (g/L) · Rd: reading in the dispersant solution (g/L) · Ms: dry mass of the subspecimen (g) · N: percent finer than the No. 10 sieve

Readings are taken at 1, 2, 4, 15, 30 and 60 min, and at 4 and 24 h, recording the temperature of each. The viscosity and density of water are computed at that temperature.

Grain-size fractions

The Unified Soil Classification System (USCS, ASTM D2487) splits the coarse particles into the fractions below; it tells silt from clay by plasticity, not size, so the curve gives them by size, as defined in ASTM D653. Each fraction's share is the difference between the percents finer at its bounds, read off the curve by interpolation on the log scale.

Fraction	Sieves	Diameter (mm)
Coarse gravel	3 in. to ¾ in.	19–75
Fine gravel	¾ in. to No. 4	4.75–19
Coarse sand	No. 4 to No. 10	2.00–4.75
Medium sand	No. 10 to No. 40	0.425–2.00
Fine sand	No. 40 to No. 200	0.075–0.425
Silt size	Finer than No. 200	0.002–0.075
Clay size	—	< 0.002

Without the hydrometer test, the curve ends at the No. 200 sieve, and what passes it is given as fines.

Curve coefficients

The diameters D10, D30 and D60, below which lie 10, 30 and 60% of the mass, sum up the shape of the curve. D10 is the effective size, linked to permeability (Hazen).

C_u = \frac{D_{60}}{D_{10}}

Cu: coefficient of uniformity · Cc: coefficient of curvature

By USCS, a gravel is well graded when Cu ≥ 4 and 1 ≤ Cc ≤ 3, and a sand when Cu ≥ 6 and 1 ≤ Cc ≤ 3; otherwise it is poorly graded. A well-graded soil has particles of every size, which fill each other's voids, giving higher density and strength; a Cc outside the range points to a missing or excess size (gap grading).

Limitations

Stokes' law assumes spherical, isolated particles; clays are platy, so hydrometer diameters are equivalent, not real.
Aggregates that survive dispersion show up as larger particles, which matters most in tropical and cemented soils.
The app uses a single control reading Rd for all readings and a linear hydrometer calibration; D7928 calls for each hydrometer's own calibration and Rd at each reading's temperature.
Grain size alone doesn't classify a soil: USCS also needs the Atterberg limits (ASTM D4318) when fines exceed 5%.

References

ASTM D6913/D6913M-17. Standard Test Methods for Particle-Size Distribution (Gradation) of Soils Using Sieve Analysis. West Conshohocken: ASTM International, 2017.
ASTM D7928-21. Standard Test Method for Particle-Size Distribution (Gradation) of Fine-Grained Soils Using the Sedimentation (Hydrometer) Analysis. West Conshohocken: ASTM International, 2021.
ASTM D2487-17. Standard Practice for Classification of Soils for Engineering Purposes (Unified Soil Classification System). West Conshohocken: ASTM International, 2017.
Holtz, R. D.; Kovacs, W. D.; Sheahan, T. C. An Introduction to Geotechnical Engineering. 2nd ed. Upper Saddle River: Pearson, 2011.
Tanaka, M. et al. Recommended table for the density of water between 0 °C and 40 °C. Metrologia, 38(4), 301–309, 2001.

Enter the sieve masses and, if run, the hydrometer readings to get the soil's grain-size curve, fractions and coefficients.

Grain-size curve

Coarse gravel: 1.0%
Fine gravel: 5.0%
Coarse sand: 4.0%
Medium sand: 24.1%
Fine sand: 38.9%
Silt size: 17.7%
Clay size: 9.3%

D10 0.00248 mmD30 0.0891 mmD60 0.331 mmCu 134Cc 9.66Poorly graded

Specimen

Total moist massgHygroscopic w%

Moist mass of the subspecimen passing No. 10g

Coarse sieving

Dry masses retained on each sieve, after washing on the No. 10 (2.00 mm) sieve.

Sieve	Retained	Finer
3 in.75 mm	g	100.0%
2 in.50 mm	g	100.0%
1½ in.37.5 mm	g	100.0%
1 in.25 mm	g	100.0%
¾ in.19 mm	g	99.0%
⅜ in.9.5 mm	g	96.8%
No. 44.75 mm	g	93.9%
No. 102 mm	g	90.0%

Hydrometer

152H hydrometer readings at the top of the meniscus. Leave blank any not taken.

Specific gravity of solids, Gs

Reading in dispersant, Rdg/LMeniscus correction, Cmg/L

Hydrometer calibration

Two points of the effective depth × reading line; 152H nominal values shown.

Reading, RDepth, Hm

Point 1g/Lcm

Point 2g/Lcm

Time	T	Reading, R
1 min	°C	g/L
2 min	°C	g/L
4 min	°C	g/L
15 min	°C	g/L
30 min	°C	g/L
1 h	°C	g/L
4 h	°C	g/L
24 h	°C	g/L

Fine sieving

Dry masses retained from the subspecimen, which passed the No. 10 sieve.

Sieve	Retained	Finer
No. 200.85 mm	g	78.9%
No. 400.425 mm	g	65.9%
No. 600.25 mm	g	53.3%
No. 1000.15 mm	g	39.6%
No. 1400.106 mm	g	33.0%
No. 2000.075 mm	g	27.0%

Calculations follow ASTM D6913 (sieve analysis) and D7928 (hydrometer analysis), with composite sieving on the No. 10 sieve and gravel and sand fractions by USCS (ASTM D2487). Water density and viscosity are computed at the temperature of each reading.