Grain size and plasticity

Two families of index tests sort soils into groups that behave alike. Grain size tests measure how much of a soil falls into each size range, which matters most for coarse soils: gravels and sands. Plasticity tests measure how a fine soil responds to changes in water content, which matters most for silts and clays. Together they are the basis of the classification systems of lesson 4, and they predict permeability, compressibility, frost susceptibility and strength well enough to guide the early decisions on any project. This lesson runs the tests on the sand, the clay and the till from the operations center.

Particle size ranges

Soil particles span an enormous range of sizes, from boulders larger than 12 in. to clay platelets smaller than 0.002 mm. The Unified Soil Classification System (USCS), described in ASTM D2487, divides them into cobbles and boulders above 3 in. (75 mm), gravel from 3 in. to the No. 4 sieve (4.75 mm), sand from the No. 4 to the No. 200 sieve (0.075 mm), and fines, silt and clay, passing the No. 200 sieve. The division between silt and clay is made by plasticity, not by size, although many references call particles smaller than 0.002 mm the clay-size fraction. The AASHTO system used for highway work places the gravel-sand boundary at the No. 10 sieve (2.0 mm) instead.

Sieve analysis

The coarse fraction is measured by sieving. A dried sample, washed first over the No. 200 sieve to remove fines that would otherwise cling to the sand grains, is shaken through a stack of sieves with openings decreasing from top to bottom. US sieves are identified by number, the number of openings per inch for the finer sizes, so the No. 4 sieve has 4.75 mm openings, the No. 40 has 0.425 mm and the No. 200 has 0.075 mm. The mass retained on each sieve is weighed, and the percentage of the total mass passing each sieve is plotted against the sieve opening on a logarithmic scale. The result is the grain size distribution curve.

The fines are measured by sedimentation in a hydrometer test. The soil is dispersed in water with a chemical that keeps the clay particles from clumping, and the density of the suspension is read at intervals. Larger particles settle faster, following Stokes' law for a sphere settling in a fluid, so the density at a given depth and time gives the percentage of particles finer than the corresponding equivalent diameter. Clay platelets are not spheres, so hydrometer sizes are equivalent diameters, not true dimensions.

Describing the curve

The shape of the curve tells how the soil was deposited and how it will behave. A steep curve means most particles are of similar size: the soil is uniform or poorly graded, typical of beach and dune sands sorted by water and wind. A flat curve spanning many sizes means the soil is well graded, so smaller particles fill the gaps between larger ones; tills and crushed aggregates are well graded. A curve with a flat step means some sizes are missing, a gap-graded soil.

Three diameters summarize the curve: \(D_{10}\), \(D_{30}\) and \(D_{60}\), the sizes at which 10, 30 and 60 % of the soil by mass is finer. \(D_{10}\) is called the effective size, because it controls permeability and capillarity. Two coefficients describe the shape:

The coefficient of uniformity \(C_u\) measures the range of sizes, and the coefficient of curvature \(C_c\) measures the shape of the curve between \(D_{10}\) and \(D_{60}\). In the USCS, a sand is well graded if \(C_u \ge 6\) and \(1 \le C_c \le 3\), and a gravel if \(C_u \ge 4\) and \(1 \le C_c \le 3\). Otherwise it is poorly graded.

Plasticity and the consistency limits

A fine soil changes its behavior as its water content changes. Very wet, it flows like a thick liquid. As it dries it becomes plastic, able to be molded without cracking. Drier still it becomes semisolid and crumbles when molded, and finally it stops shrinking as it dries and is solid. Around 1911 the Swedish scientist Albert Atterberg defined the water contents at the boundaries between these states, and Arthur Casagrande later standardized the tests. They are called the Atterberg limits or consistency limits:

• The liquid limit (LL) is the water content at which the soil passes from plastic to liquid behavior.
• The plastic limit (PL) is the water content at which the soil can no longer be rolled into a 3 mm (1/8 in.) thread without crumbling.
• The shrinkage limit (SL) is the water content below which further drying causes no further loss of volume.

Measuring the limits

In the Casagrande method, a pat of soil paste is placed in a brass cup, a groove is cut through it with a standard tool, and the cup is dropped 10 mm onto a hard rubber base by turning a crank. The number of blows needed to close the groove over 13 mm (1/2 in.) is recorded. The test is repeated at several water contents, and the liquid limit is the water content that corresponds to 25 blows on a straight line fitted to water content against the logarithm of the number of blows, called the flow curve. Many laboratories outside North America use the fall cone instead, in which a standard cone drops into the paste; the liquid limit corresponds to a fixed penetration.

The plastic limit is found by rolling a ball of soil on a glass plate into a thread. When the thread crumbles at a diameter of 3 mm (1/8 in.), its water content is the plastic limit. The test is simple but operator-dependent, and experienced technicians agree within a couple of percentage points.

From the limits come the indices that describe the soil:

The plasticity index \(PI\) is the range of water content over which the soil is plastic. The liquidity index \(LI\) places the natural water content within that range: 0 at the plastic limit, 1 at the liquid limit. A soil with \(LI\) near 1 or above is soft and sensitive and may lose much of its strength when remolded; one near 0 is stiff. Skempton's activity \(A\) relates plasticity to the amount of clay and indicates the clay mineral: below about 0.75 is inactive, typical of kaolinite and illite; above 1.25 is active, typical of smectite, which swells.

Running the tests on the site soils

Owen sends samples of the three main soils from B-3 to the laboratory: the sand at 5 ft, the clay at 20 ft and the till at 40 ft. Sofia's script interpolates the characteristic diameters, splits each soil into gravel, sand and fines, fits the flow curve and computes the indices. It also lists the sand curve by US sieve number, the way the laboratory reports it.

# Grain size and plasticity of the three main soils at the operations center
from math import log10

def d_at(curve, p):
    """Diameter (mm) at p % passing, by interpolation on a log scale."""
    pts = sorted(curve)                       # (diameter, % passing), finest first
    for (d1, p1), (d2, p2) in zip(pts, pts[1:]):
        if p1 <= p <= p2:
            return 10 ** (log10(d1) + (p - p1) / (p2 - p1) * (log10(d2) - log10(d1)))
    return None

def pct_finer(curve, d):
    pts = sorted(curve)
    for (d1, p1), (d2, p2) in zip(pts, pts[1:]):
        if d1 <= d <= d2:
            return p1 + (log10(d) - log10(d1)) / (log10(d2) - log10(d1)) * (p2 - p1)
    return None

soils = {
 "sand (B-3, 5 ft)": [(9.5, 100), (4.75, 98), (2.0, 93), (0.85, 80), (0.425, 55), (0.25, 30),
                        (0.15, 15), (0.075, 7), (0.02, 3), (0.005, 1)],
 "till (B-3, 40 ft)":  [(75, 100), (37.5, 94), (19, 86), (9.5, 77), (4.75, 69), (2.0, 61), (0.85, 53),
                        (0.425, 46), (0.25, 41), (0.15, 37), (0.075, 32), (0.02, 20), (0.005, 11), (0.002, 7)],
 "clay (B-3, 20 ft)":   [(0.425, 100), (0.075, 99), (0.05, 97), (0.02, 85), (0.01, 70), (0.005, 52), (0.002, 38)],
}
for name, c in soils.items():
    gravel = 100 - pct_finer(c, 4.75) if c[0][0] > 4.75 else 0.0
    fines = pct_finer(c, 0.075)
    print(f"{name}: gravel {gravel:.0f} %, sand {100 - gravel - fines:.0f} %, fines {fines:.0f} %, "
          f"finer than 0.02 mm {pct_finer(c, 0.02):.0f} %")
    d10, d30, d60 = (d_at(c, p) for p in (10, 30, 60))
    if d10:
        print(f"   D10 = {d10:.3f}  D30 = {d30:.3f}  D60 = {d60:.3f} mm  "
              f"Cu = {d60/d10:.1f}  Cc = {d30**2/(d10*d60):.2f}")

# Liquid limit of the clay from four Casagrande cup points: fit w against log N
pts = [(34, 43.1), (27, 44.6), (21, 46.0), (15, 48.1)]
xs = [log10(n) for n, _ in pts]; ys = [w for _, w in pts]
xm, ym = sum(xs) / 4, sum(ys) / 4
slope = sum((x - xm) * (y - ym) for x, y in zip(xs, ys)) / sum((x - xm) ** 2 for x in xs)
LL = ym + slope * (log10(25) - xm)
PL, w_nat, clay2um = 22.0, 38.5, 38.0
PI = LL - PL
print(f"\nFlow index = {-slope:.1f} per log cycle; LL = {LL:.1f} %, PL = {PL:.0f} %, PI = {PI:.1f} %")
print(f"Liquidity index = {(w_nat - PL)/PI:.2f}; activity = PI / clay fraction = {PI/clay2um:.2f}")

# The same sand curve expressed by US standard sieve number
for sieve, d in (("No. 4", 4.75), ("No. 10", 2.0), ("No. 20", 0.85), ("No. 40", 0.425), ("No. 60", 0.25), ("No. 100", 0.15), ("No. 200", 0.075)):
    print(f"{sieve:8s} {d:6.3f} mm  {pct_finer(soils['sand (B-3, 5 ft)'], d):5.0f} % passing")

sand (B-3, 5 ft): gravel 2 %, sand 91 %, fines 7 %, finer than 0.02 mm 3 %
   D10 = 0.097  D30 = 0.250  D60 = 0.488 mm  Cu = 5.0  Cc = 1.32
till (B-3, 40 ft): gravel 31 %, sand 37 %, fines 32 %, finer than 0.02 mm 20 %
   D10 = 0.004  D30 = 0.060  D60 = 1.797 mm  Cu = 452.0  Cc = 0.51
clay (B-3, 20 ft): gravel 0 %, sand 1 %, fines 99 %, finer than 0.02 mm 85 %

Flow index = 14.0 per log cycle; LL = 45.0 %, PL = 22 %, PI = 23.0 %
Liquidity index = 0.72; activity = PI / clay fraction = 0.61
No. 4     4.750 mm     98 % passing
No. 10    2.000 mm     93 % passing
No. 20    0.850 mm     80 % passing
No. 40    0.425 mm     55 % passing
No. 60    0.250 mm     30 % passing
No. 100   0.150 mm     15 % passing
No. 200   0.075 mm      7 % passing

The till is the opposite of the sand: its curve is long and flat, with \(C_u\) over 400, because the ice mixed everything from clay to cobbles. Its \(C_c\) of 0.5 is outside the range for a well-graded soil only because the curve is so flat in the middle; in practice it is described as broadly graded. Its 32 % fines will control its behavior, and the classification in lesson 4 depends on their plasticity.

Plasticity and engineering behavior

Plasticity is a proxy for many properties that are slow or costly to measure directly. More plastic clays are more compressible, less permeable, weaker in drained shear and more prone to shrinkage and swelling. Empirical correlations exploit this: lesson 10 estimates the compression index of the site clay from its liquid limit, and drained friction angles of clays fall as the plasticity index rises. These correlations are no substitute for tests on undisturbed samples, but they check the tests and fill in gaps.

Frost susceptibility

Ice lenses form in soils that are fine enough to draw water upward by capillarity toward a freezing front but permeable enough to supply it quickly. Clean sands and gravels are permeable but have weak suction, so they freeze in place without heaving. Clays have strong suction but transmit water too slowly. Silts and silty sands are the worst: they heave and then lose strength as the ice melts in spring. Casagrande's criterion, still used as a first screen, flags a soil as frost-susceptible when more than about 3 % of a well-graded soil, or about 10 % of a uniform soil, is finer than 0.02 mm.

The site sand has about 3 % finer than 0.02 mm. As a uniform sand it falls below Casagrande's threshold and is at most slightly frost-susceptible; it can be used for backfill around footings. The till, with about 20 % finer than 0.02 mm, is highly frost-susceptible and must not be used against foundation walls or under slabs in unheated areas. The US Army Corps of Engineers frost groups and the specifications of state DOTs, including Minnesota's, take a similar approach, limiting the fines in granular base and backfill.

Practical points

• Always wash the sample over the finest sieve before dry sieving; dry sieving alone underestimates the fines.
• Run the Atterberg limits on the fraction passing the 0.425 mm (No. 40) sieve, as the standards require.
• Do not oven-dry organic or sensitive clays before the limits; drying changes them irreversibly. Start from the natural water content.
• Compare the natural water content with the limits on every sample. A clay near its liquid limit needs special care in excavation.