About the Particle Size Distribution Calculator

This Particle Size Distribution Calculator (D10/D50/D90) is an essential tool for geotechnical engineers, material scientists, and geologists. It analyzes data from a sieve analysis to characterize the grain size distribution of granular materials like soil and aggregates. By calculating key parameters, it helps in classifying materials and predicting their engineering behavior.

What This Calculator Does

The calculator processes pairs of particle sizes and their corresponding percentage values (passing, retained, or frequency) to generate a complete particle size distribution curve. It uses log-linear interpolation—the standard method for this type of analysis—to determine specific particle diameters at given cumulative percentages.

Key outputs include:

• Characteristic Diameters: D10, D30, D50, D60, and D90, which represent the particle sizes for which 10%, 30%, 50%, etc., of the material by weight is finer.
• Gradation Coefficients: The Coefficient of Uniformity (Cu) and the Coefficient of Curvature (Cc), which are crucial for soil classification systems like the Unified Soil Classification System (USCS).
• Distribution Span: A measure of the width of the size distribution relative to the median particle size.
• Gradation Curve Plot: A semi-logarithmic graph visualizing the distribution of particle sizes.

When to Use It

This tool is valuable in various professional and academic contexts:

• Geotechnical Engineering: For classifying soils (e.g., as well-graded or poorly-graded), assessing soil permeability, frost susceptibility, and shear strength.
• Construction Materials: To ensure aggregates for concrete or asphalt meet specific gradation requirements for optimal strength and durability.
• Environmental Science: To characterize sediment in rivers or on coastlines, which affects transport and ecological habitats.
• Powder Metallurgy & Pharmaceuticals: To control the size distribution of powders, which influences compaction, flowability, and dissolution rates.

Inputs Explained

• Particle Size & Percent Value: The core data from a sieve analysis. You must provide at least two data points.
  • Particle Size: The opening size of the sieve (e.g., in mm, μm, or inches).
  • Percent Value: The corresponding percentage, which can be entered in one of three types.
• Input Percent Type:
  • Cumulative % Passing: The percentage of material by weight that passes through a given sieve. This is the most common format. The values should increase as sieve size decreases.
  • Cumulative % Retained: The percentage of material by weight that is retained on a given sieve. The calculator will convert this to % Passing by subtracting from 100.
  • Frequency %: The percentage of material retained on an individual sieve (not cumulative). The calculator will first sort the data by size and then compute the cumulative passing percentages.
• Custom D-Value: Allows you to calculate the particle size for any custom percentage (e.g., entering "84" will calculate D84).

Results Explained

• D10 (Effective Size): The particle diameter at which 10% of the sample is finer. It is a critical parameter for estimating hydraulic conductivity and assessing soil liquefaction potential.
• D30, D60, D90: Diameters at which 30%, 60%, and 90% of the sample is finer, respectively. These are used to calculate the gradation coefficients.
• D50 (Median Diameter): The particle diameter at which 50% of the sample is finer and 50% is coarser. It represents the average particle size.
• Coefficient of Uniformity (Cu): Calculated as Cu = D60 / D10. It indicates the range of particle sizes in the material. A high Cu value (>4 for gravels, >6 for sands) suggests a wide range of sizes (well-graded).
• Coefficient of Curvature (Cc): Calculated as Cc = (D30)² / (D10 * D60). It describes the smoothness of the gradation curve. For a well-graded soil, Cc should be between 1 and 3.
• Span: Calculated as Span = (D90 - D10) / D50. It provides a non-dimensional measure of the distribution's width. A smaller span indicates a more uniform particle size.

Formula / Method

The calculator determines D-values (like D10, D50, etc.) that fall between two measured data points using log-linear interpolation. Given two points (D1, P1) and (D2, P2), where D is the particle size and P is the cumulative percent passing, the size Dx for a target percentage Px is found with the formula:

log10(Dx) = log10(D1) + [ (log10(D2) - log10(D1)) * (Px - P1) / (P2 - P1) ]

Once the required D-values are found, the coefficients are calculated as defined above.

Step-by-Step Example

Let's find D50 and D60 from the following % Passing data to eventually calculate Cu.

1. Find D60:

The target percentage Px = 60% falls between P1 = 49.0% (at D1 = 2.36 mm) and P2 = 64.0% (at D2 = 4.75 mm).

log10(D60) = log10(2.36) + [ (log10(4.75) - log10(2.36)) * (60 - 49.0) / (64.0 - 49.0) ]

log10(D60) = 0.3729 + [ (0.6767 - 0.3729) * 11 / 15 ]

log10(D60) = 0.3729 + [ 0.3038 * 0.7333 ] = 0.5957

D60 = 10^0.5957 = 3.94 mm

2. Find D50:

The target percentage Px = 50% also falls between the same two points.

log10(D50) = log10(2.36) + [ (log10(4.75) - log10(2.36)) * (50 - 49.0) / (64.0 - 49.0) ]

log10(D50) = 0.3729 + [ 0.3038 * 1 / 15 ] = 0.3932

D50 = 10^0.3932 = 2.47 mm

The calculator repeats this process for D10 and D30 to find Cu and Cc.

Tips + Common Errors

• Insufficient Data: You need at least two data points to perform interpolation. More data points across a wider range will yield a more accurate curve.
• Extrapolation Error: The calculator cannot determine D-values that fall outside the range of your provided percentage data. For example, if your minimum % passing is 12%, it cannot calculate D10. Ensure your data brackets the percentages you need.
• Incorrect Data Type: Double-check whether your data is cumulative % passing, cumulative % retained, or frequency %. Selecting the wrong type will lead to incorrect results.
• Duplicate Sizes: Ensure each particle size entry is unique. Duplicate sizes can cause errors in calculation and plotting.
• Data Sorting: While the calculator sorts data automatically, it's good practice to enter it logically (e.g., from largest to smallest particle size).

Frequently Asked Questions (FAQs)

1. What is the difference between D10, D50, and D90?
   They are statistical points on the distribution curve. D10 (effective size) represents the finest 10% of particles, D50 (median size) is the midpoint of the distribution, and D90 represents the coarsest 10% of particles.
2. How do I interpret the Coefficient of Uniformity (Cu)?
   Cu measures the uniformity of particle sizes. A low Cu (e.g., < 4) means the particles are of similar size (poorly-graded or uniform). A high Cu (> 6 for sands, > 4 for gravels) indicates a wide range of particle sizes (well-graded).
3. What does a Coefficient of Curvature (Cc) between 1 and 3 mean?
   A Cc value between 1 and 3 indicates a smooth, continuous gradation curve, representing a good distribution of particle sizes. Values outside this range suggest a "gap-graded" soil, where certain intermediate sizes are missing.
4. Can this calculator be used for official USCS soil classification?
   This calculator provides the necessary parameters (Cu, Cc) for classification according to the Unified Soil Classification System (USCS). However, a full classification also requires information about fines content (% passing the No. 200 sieve) and plasticity (Atterberg limits).
5. What happens if the calculator shows "N/A" for Cu or Cc?
   This means that D10, D30, or D60 could not be calculated because the required percentage (10%, 30%, or 60%) fell outside the range of your input data. You must provide data points that bracket these percentages.
6. Why does the graph use a logarithmic scale for particle size?
   Particle sizes in soils can span several orders of magnitude (from microns to centimeters). A logarithmic scale allows the entire range to be displayed clearly on a single graph, giving equal visual importance to each size range.
7. What is the difference between poorly-graded and well-graded soil?
   A well-graded soil has a good representation of all particle sizes, leading to better compaction and higher strength. A poorly-graded (or uniformly-graded) soil consists mainly of particles of the same size, resulting in more void space and lower stability.
8. Is it better to use manual entry or paste data?
   For a small number of data points, manual entry is fine. For larger datasets from a lab report or spreadsheet, the paste data function is faster and reduces the risk of typing errors.

References

1. ASTM D2487-17: Standard Practice for Classification of Soils for Engineering Purposes (Unified Soil Classification System). ASTM International.
2. ASTM D422-63(2007): Standard Test Method for Particle-Size Analysis of Soils (Withdrawn 2016, but historically significant). ASTM International.
3. Das, B. M., & Sobhan, K. (2013). Principles of Geotechnical Engineering. Cengage Learning. (See Chapter 3: Weight-Volume Relationships, Plasticity, and Soil Classification).
4. U.S. Army Corps of Engineers. (1986). Engineering and Design - Laboratory Soils Testing (EM 1110-2-1906). (See Appendix III: Grain-Size Analysis).