If the water content of a thick soil-water mixture is gradually reduced, the mixture passes from liquid state to plastic state, then to semi-solid state, and finally to solid state. The water contents at which the soil changes from one state to another are called Atterberg limits or consistency limits.
Liquid Limit (LL): water content at which soil changes from liquid state to plastic state.
Plastic Limit (PL): water content below which soil stops behaving as a plastic material.
Shrinkage Limit (SL): smallest water content at which the soil is saturated, or the maximum water content at which further loss of water no longer causes volume decrease.
Plasticity and Consistency Indices
These indices compare the natural water content with the Atterberg limits.
$$PI = LL - PL$$
$$LI = \frac{w - PL}{PI}$$
$$I_c = \frac{LL - w}{PI}$$
$$I_c + LI = 1$$
When $LI < 0$, the soil behaves as a brittle solid. When $0 < LI < 1$, it is in the plastic range. When $LI > 1$, it is in a liquid state.
Shrinkage and Strength Indices
Shrinkage and strength-related indices are often used to describe volume change and clay behavior.
$$SI = PL - SL$$
$$VS = w - SL$$
$$SR = \frac{M_d}{V_d\rho_w}$$
$$SL = w_i - \frac{(V_i - V_d)\rho_w}{M_d}(100)$$
$$G_s = \frac{1}{\frac{1}{SR} - \frac{SL}{100}}$$
Here, $M_d$ is dry mass, $V_d$ is dry volume, $V_i$ is initial wet volume, and $\rho_w$ is the density of water.
Flow, Toughness, Sensitivity, and Activity
The flow index comes from the slope of the flow curve in the liquid limit test. Toughness index relates plasticity to flow. Sensitivity compares undisturbed strength with remolded strength at the same water content.
In the liquid limit device, a soil paste is placed in a brass cup and grooved with a standard tool. The cup is dropped repeatedly from a height of 10 mm until the groove closes by 12.7 mm. The water content corresponding to 25 blows is the liquid limit.
With the cone penetrometer method, the liquid limit is commonly taken as the water content at 20 mm cone penetration.
Plastic Limit and Shrinkage Limit Tests
The plastic limit is found by rolling soil into threads until it crumbles at about 3 mm diameter. The plastic limit is the average water content of the soil samples that crumble at this thread size.
The shrinkage limit test uses a dish filled with wet soil. The wet mass and volume are recorded, then the sample is oven dried and its dry mass and dry volume are measured. The shrinkage limit is the water content at which further water loss causes no further volume change.
Problem: Toughness Index
A soil has a liquid limit of 25 percent and a flow index of 12.5 percent. If the plastic limit is 15 percent, compute the toughness index.
$$PI = LL - PL$$
$$PI = 25 - 15 = 10$$
$$I_t = \frac{PI}{I_f} = \frac{10}{12.5} = 0.80$$
Answer: $I_t = 0.80$.
Problem: Relative Consistency
A soil has a liquid limit of 32 percent and a plastic limit of 15 percent. Compute the relative consistency if the water content of the soil in its natural condition is 20 percent.
$$PI = 32 - 15 = 17$$
$$I_c = \frac{LL - w}{PI}$$
$$I_c = \frac{32 - 20}{17} = 0.706$$
Answer: Relative consistency is 0.706.
Problem: Plasticity Index, Sensitivity, and Activity
From the given properties of a soil sample: liquid limit = 61 percent, plastic limit = 20 percent, undisturbed shear strength = 18 kN/m2, clay-size fraction finer than 0.002 mm = 79 percent, and remolded shear strength = 1.3 kN/m2. Compute the plasticity index, sensitivity, and activity.
Problem: Relative Consistency from Field Water Content
A soil in its natural condition in the field has a water content of 22 percent. If it has a liquid limit of 25 percent and a plastic limit of 15 percent, compute the relative consistency.
$$PI = 25 - 15 = 10$$
$$I_c = \frac{25 - 22}{10} = 0.30$$
Answer: Relative consistency is 0.30.
Problem: Consistency Index of Clay
The Atterberg limits of a clay are as follows: liquid limit = 60 percent, plastic limit = 45 percent, natural moisture content = 50 percent, specific gravity of soil solids = 2.70, and shrinkage limit = 25 percent. Compute the consistency index.
$$PI = 60 - 45 = 15$$
$$I_c = \frac{60 - 50}{15} = 0.667$$
Answer: $I_c = 0.667$.
Problem: Consistency Index from Atterberg Limits
A clay has the following Atterberg limits: liquid limit = 63 percent, plastic limit = 23 percent, and water content = 38 percent. Compute the consistency index.
$$PI = 63 - 23 = 40$$
$$I_c = \frac{63 - 38}{40} = 0.625$$
Answer: $I_c = 0.625$.
Problem: Liquidity Index from Limit Test
From a limit test, the weight of wet soil is 9 g and the weight of dry soil is 7 g. The natural water content of the soil is 38 percent. Compute the liquidity index if the soil has a liquid limit of 64 percent.
The liquidity index locates the natural water content on a scale between the plastic limit and the liquid limit. A value near 1 means the soil is nearly as soft as slurry, compressible, and sensitive to disturbance. A value near 0 means the soil is stiff and close to the plastic limit. A negative value means the soil has dried below the plastic limit and behaves as a brittle solid that may crack if remolded.
The consistency index $I_c$ is the mirror of the liquidity index: $I_c + LI = 1$. High consistency index indicates stiff soil; low consistency index indicates soft soil. Engineers use both indices to estimate undrained shear strength and sensitivity without running a full triaxial test.
Problem: Shrinkage Limit from Laboratory Measurements
A wet soil sample has a mass of 75.4 g and a volume of 39.5 cm3. After oven drying, the dry mass is 52.0 g and the dry volume is 26.8 cm3. Compute the initial water content and the shrinkage limit.
In a liquid limit test using the Casagrande cup, two readings are recorded: at 18 blows the water content is 52.4 percent, and at 32 blows the water content is 47.8 percent. Determine the flow index and estimate the liquid limit.
A soil sample in a shrinkage dish has a dry mass of 45.2 g and a dry volume of 27.5 cm3. The shrinkage limit is 18 percent. Compute the shrinkage ratio and the specific gravity of the soil solids.
Problem: Full Atterberg Analysis and State Identification
A clay soil has a liquid limit of 55 percent, a plastic limit of 25 percent, and a shrinkage limit of 14 percent. The natural water content in the field is 48 percent. Compute the plasticity index, liquidity index, consistency index, and shrinkage index. State whether the soil is in a liquid, plastic, or solid condition.
$$PI = LL-PL=55-25=30$$
$$LI = \frac{w-PL}{PI}=\frac{48-25}{30}=0.767$$
$$I_c = \frac{LL-w}{PI}=\frac{55-48}{30}=0.233$$
$$SI = PL-SL=25-14=11$$
Since $0 < LI < 1$, the soil is in the plastic range. Because $LI=0.767$ is closer to 1.0, the clay is on the soft end of its plastic state.
Answer: $PI=30$, $LI=0.767$, $I_c=0.233$, $SI=11$; the soil is in the plastic state (soft).
Problem: Water to Remove to Reach the Plastic Limit
A clay has a liquid limit of 65 percent, a plasticity index of 30, and a natural water content of 40 percent. A soil engineer wants to rework the clay so its water content equals the plastic limit. How much water must be removed per kilogram of dry soil?
$$PL = LL-PI=65-30=35\%$$
The natural water content is 40 percent. The target is 35 percent. The amount of water per kilogram of dry soil to remove is:
