Chapter 4 Geotechnical Engineering: Particle size distribution

1. Introduction: Particle size distribution

2. Simulation: PSD

3. Self-Assessment

1. Introduction

Fig. 15 **Figure 4.1 **: Particle Size Distribution.

🧱 Particle Size and Distribution in Geotechnical Properties

Particle Size Distribution (PSD) refers to the proportion of different grain sizes present in a soil sample. It is typically determined through:

• Sieve analysis for coarse-grained soils (sand, gravel)

• Hydrometer analysis for fine-grained soils (silt, clay): Based on sedimentation and Stokes’ Law; - Suitable for fine-grained soils (silt and clay)- Measures settling velocity in a fluid column

The results are plotted as a gradation curve, showing the percentage of material finer than a given size.

🔬 Alternative Methods for Particle Size Distribution (PSD)

Beyond traditional sieve analysis, several advanced techniques are used to characterize particle size across a wide range of materials and scales:

• Laser Diffraction (LD)

  • Uses light scattering to estimate particle size - Covers ~0.01 µm to 2000 µm range - Assumes spherical particles; rapid and automated

• Dynamic Light Scattering (DLS)

  • Measures Brownian motion of particles in suspension - Ideal for nanoparticles (0.3 nm to ~10 µm) - Provides hydrodynamic diameter

• Dynamic Image Analysis (DIA)

  • Captures real-time images of particles in motion - Measures both size and shape - Useful for multimodal and irregular distributions

• Scanning Electron Microscopy (SEM)

  • Direct imaging of particle morphology and size - Suitable for particles >10 nm - Offers detailed surface texture analysis

• Coulter Counter (Electrical Sensing Zone Method)

  • Detects changes in electrical resistance as particles pass through an orifice - Effective for sizing and counting particles in suspension

These methods vary in resolution, cost, and assumptions about particle shape, making them complementary tools depending on the material and engineering context.

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📐 Key Parameters from PSD

Parameter	Description
D10	Diameter at which 10% of soil is finer (effective size)
D30, D60	Diameters at 30% and 60% finer; used in uniformity and gradation calculations
Cu (Uniformity Coefficient)	( Cu = \frac{D60}{D10} ); indicates soil gradation
Cc (Coefficient of Curvature)	( Cc = \frac{D30^2}{D10 \cdot D60} ); indicates shape of curve

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🧠 Why Particle Size Distribution Is Important

1. Classification

• Determines whether soil is coarse-grained (sand, gravel) or fine-grained (silt, clay)

• Used in systems like USCS and AASHTO

2. Permeability

• Well-graded soils (wide range of sizes) have lower permeability

• Uniform soils (similar sizes) allow faster drainage

3. Compaction and Density

• PSD affects maximum dry density and optimum moisture content

• Well-graded soils compact better and achieve higher strength

4. Shear Strength

• Coarse particles provide frictional resistance

• Fines influence cohesion and plasticity

5. Frost Susceptibility

• Fine soils (especially silts) are more prone to frost heave

6. Filter Design and Drainage

• PSD ensures proper filter compatibility to prevent piping and clogging

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📊 Role of Particle Size in Geotechnical Properties

Property	Influence of Particle Size
Strength	Larger particles increase friction; fines add cohesion
Compressibility	Fine soils compress more under load
Permeability	Coarse soils drain faster; fines retain water
Plasticity	Controlled by clay content and mineralogy
Stability	Well-graded soils resist erosion and settlement

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🧠 Conceptual Insight

Particle size distribution is the foundation of soil classification and behavior prediction —
it governs how soil responds to loading, drainage, compaction, and environmental changes.

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References

[Das, 2010] provides a clear and pedagogically rich explanation of particle size distribution (PSD). Furthermore, the three ASTM standards for particle size distribution (PSD) testing provide complementary methods for analyzing soil gradation. ASTM D422 [ASTM International, 1963], now withdrawn, historically combined sieve and hydrometer techniques to assess both coarse and fine fractions in a single procedure. It has since been replaced by two more specialized standards: ASTM D6913 [ASTM International, 2004], which covers sieve analysis for coarse-grained soils such as sand and gravel, and ASTM D7928 {cite: p}ASTM_D7928, which details hydrometer analysis for fine-grained soils like silt and clay. D6913 outlines procedures for dry sieving, including calculations for the coefficient of uniformity and curvature, while D7928 applies sedimentation principles based on Stokes’ Law, incorporating corrections for temperature and dispersant effects. Together, D6913 and D7928 form the current ASTM framework for accurately characterizing soil gradation across the full particle size spectrum.

2. Simulation

Interactive Particle Size Distribution (PSD) Tool

This notebook provides an interactive tool for analyzing Particle Size Distribution (PSD) in soil mechanics. It is designed for educational use and includes:

This tool:

• Computes key PSD parameters: D10, D30, D60

• Calculates Uniformity Coefficient (Cu) and Curvature Coefficient (Cc)

• Classifies soil gradation (e.g., GW, SW, GP, SP) based on empirical criteria

• Visualizes the gradation curve using Plotly

• Interprets soil type based on fines content and PSD shape

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🔧 Computational Steps

1. User Inputs:

   • Sieve Sizes (mm): Grain diameters used in sieve analysis

   • Percent Passing (%): Cumulative percentage of soil passing each sieve

2. Calculations:

   • Interpolates D10, D30, D60 from the gradation curve

   • Computes: \( Cu = \frac{D60}{D10}, \quad Cc = \frac{D30^2}{D10 \cdot D60} \)

   • Determines fines content (percent passing 0.075 mm)

   • Applies classification rules based on Cu, Cc, and fines

3. Visualization:

   • Plots PSD curve with D-values highlighted

   • Uses log scale and reversed axes for standard soil gradation format

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📊 How to Interpret Inputs

Input Field	Meaning
Sieve Sizes (mm)	Grain sizes used in sieve analysis
Percent Passing (%)	Cumulative percent of soil finer than each sieve size

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📋 How to Interpret Outputs

Output Parameter	Interpretation
D10, D30, D60	Key grain sizes used in classification and permeability estimation
Cu	Indicates soil gradation; Cu ≥ 4 (gravel), Cu ≥ 6 (sand) = well-graded
Cc	Indicates curve shape; 1 ≤ Cc ≤ 3 = well-graded
Percent Fines	Helps distinguish coarse vs. fine-grained soils
Classification	Soil type based on gradation and fines (e.g., GW, SW, SP, SC, SM)
PSD Curve	Visual representation of soil gradation

---

🧠 Conceptual Insight

This tool links grain size distribution to soil classification and engineering behavior —
helping geotechnical engineers assess drainage, compaction, and stability characteristics.

import numpy as np
import plotly.graph_objects as go
import ipywidgets as widgets
from IPython.display import display, Markdown

def compute_d_values(sieve_sizes, percent_passing):
    D10 = np.interp(10, percent_passing[::-1], sieve_sizes[::-1])
    D30 = np.interp(30, percent_passing[::-1], sieve_sizes[::-1])
    D60 = np.interp(60, percent_passing[::-1], sieve_sizes[::-1])
    return D10, D30, D60

def plot_psd(sieve_sizes, percent_passing, D10, D30, D60):
    fig = go.Figure()
    fig.add_trace(go.Scatter(x=sieve_sizes, y=percent_passing, mode='lines+markers', name='PSD Curve'))
    fig.add_trace(go.Scatter(x=[D10], y=[10], mode='markers+text', name='D10',
                             text=["D10"], textposition="top center", marker=dict(size=10, color='red')))
    fig.add_trace(go.Scatter(x=[D30], y=[30], mode='markers+text', name='D30',
                             text=["D30"], textposition="top center", marker=dict(size=10, color='green')))
    fig.add_trace(go.Scatter(x=[D60], y=[60], mode='markers+text', name='D60',
                             text=["D60"], textposition="top center", marker=dict(size=10, color='blue')))
    fig.update_layout(title='Particle Size Distribution Curve',
                      xaxis_title='Sieve Size (mm)',
                      yaxis_title='Percent Passing (%)',
                      xaxis_type='log',
                      xaxis=dict(autorange='reversed'),
                      yaxis=dict(autorange='reversed'),
                      width=800, height=500)
    fig.show()

def classify_soil_by_gradation(Cu, Cc):
    if Cu >= 4 and 1 <= Cc <= 3:
        return "Well-graded gravel (GW)"
    elif Cu >= 6 and 1 <= Cc <= 3:
        return "Well-graded sand (SW)"
    else:
        return "Poorly graded soil (GP/SP)"

sieve_input = widgets.Text(
    value='4.75, 2.0, 1.0, 0.425, 0.25, 0.075',
    description='Sieve Sizes (mm):',
    layout=widgets.Layout(width='100%'),
    style={'description_width': 'initial'}
)

passing_input = widgets.Text(
    value='100, 90, 70, 50, 30, 10',
    description='Percent Passing (%):',
    layout=widgets.Layout(width='100%'),
    style={'description_width': 'initial'}
)

output = widgets.Output()

def update_plot(change=None):
    output.clear_output()
    with output:
        try:
            sieve_sizes = np.array([float(x.strip()) for x in sieve_input.value.split(',')])
            percent_passing = np.array([float(x.strip()) for x in passing_input.value.split(',')])
            if len(sieve_sizes) != len(percent_passing):
                display(Markdown("❌ **Error:** Number of sieve sizes and percent passing values must match."))
                return
            sorted_indices = np.argsort(sieve_sizes)[::-1]
            sieve_sizes = sieve_sizes[sorted_indices]
            percent_passing = percent_passing[sorted_indices]
            D10, D30, D60 = compute_d_values(sieve_sizes, percent_passing)
            Cu = D60 / D10 if D10 != 0 else np.nan
            Cc = (D30**2) / (D10 * D60) if D10 != 0 and D60 != 0 else np.nan
            classification = classify_soil_by_gradation(Cu, Cc)
            display(Markdown(f"### 📐 Calculated Parameters"))
            display(Markdown(f"- **D10:** {D10:.3f} mm  - **D30:** {D30:.3f} mm  - **D60:** {D60:.3f} mm"))
            display(Markdown(f"- **Coefficient of Uniformity (Cu):** {Cu:.2f}  - **Coefficient of Curvature (Cc):** {Cc:.2f}"))
            display(Markdown(f"### 🧠 Soil Gradation Hint: **{classification}**"))
            plot_psd(sieve_sizes, percent_passing, D10, D30, D60)
        except Exception as e:
            display(Markdown(f"❌ **Error:** {e}"))

sieve_input.observe(update_plot, names='value')
passing_input.observe(update_plot, names='value')

display(widgets.VBox([sieve_input, passing_input]), output)
update_plot()


import numpy as np

def classify_soil_from_psd(sieve_sizes, percent_passing):
    sorted_indices = np.argsort(sieve_sizes)[::-1]
    sieve_sizes = np.array(sieve_sizes)[sorted_indices]
    percent_passing = np.array(percent_passing)[sorted_indices]

    D10 = np.interp(10, percent_passing[::-1], sieve_sizes[::-1])
    D30 = np.interp(30, percent_passing[::-1], sieve_sizes[::-1])
    D60 = np.interp(60, percent_passing[::-1], sieve_sizes[::-1])

    Cu = D60 / D10 if D10 != 0 else np.nan
    Cc = (D30 ** 2) / (D10 * D60) if D10 != 0 and D60 != 0 else np.nan

    fines_index = np.where(np.array(sieve_sizes) <= 0.075)[0]
    percent_fines = percent_passing[fines_index[0]] if fines_index.size > 0 else 0

    if percent_fines > 50:
        classification = "Fine-grained soil (Use Atterberg limits for further classification)"
    elif percent_fines > 12:
        classification = "Coarse-grained soil with fines (e.g., SC, SM, GC, GM)"
    else:
        if D10 == 0 or D60 == 0:
            classification = "Insufficient data for classification"
        elif Cu >= 4 and 1 <= Cc <= 3:
            classification = "Well-graded gravel (GW)" if sieve_sizes[0] > 4.75 else "Well-graded sand (SW)"
        else:
            classification = "Poorly graded gravel (GP)" if sieve_sizes[0] > 4.75 else "Poorly graded sand (SP)"

    return {
        'D10 (mm)': round(D10, 3),
        'D30 (mm)': round(D30, 3),
        'D60 (mm)': round(D60, 3),
        'Cu': round(Cu, 2),
        'Cc': round(Cc, 2),
        'Percent Fines': round(percent_fines, 1),
        'Classification': classification
    }

# Example dataset
sieve_sizes = [4.75, 2.0, 1.0, 0.425, 0.25, 0.075]
percent_passing = [100, 90, 70, 50, 30, 10]

# Run classification
result = classify_soil_from_psd(sieve_sizes, percent_passing)

# Display results
for key, value in result.items():
    print(f"{key}: {value}")

D10 (mm): 0.075
D30 (mm): 0.25
D60 (mm): 0.712
Cu: 9.5
Cc: 1.17
Percent Fines: 10
Classification: Well-graded sand (SW)

3. Self-Assessment

Reflective Questions

1. How does particle size distribution affect the properties of materials in different industries?

2. What are the advantages and limitations of different particle size measurement techniques?

3. How can understanding particle size distribution improve product quality and performance?

4. What challenges might arise when interpreting particle size distribution data?

Conceptual Questions

1. Define particle size distribution and explain its significance in material science.

2. Describe the process of laser diffraction and how it is used to measure particle size.

3. Explain the importance of D10, D30, and D60 values in particle size distribution analysis.

4. What is the difference between well-graded and poorly graded soils? How are these classifications determined?

Quiz

import numpy as np
import ipywidgets as widgets
from IPython.display import display, Markdown

# Quiz Questions
questions = [
    {
        "question": "What does D10 represent in particle size distribution?",
        "options": ["The diameter at which 10% of the sample's mass is finer", "The diameter at which 10% of the sample's mass is coarser", "The average particle size", "The largest particle size"],
        "answer": "The diameter at which 10% of the sample's mass is finer"
    },
    {
        "question": "Which technique uses laser light scattering to measure particle size?",
        "options": ["Dynamic Light Scattering", "Laser Diffraction", "Image Analysis", "Centrifugal Sedimentation"],
        "answer": "Laser Diffraction"
    },
    {
        "question": "What is the Coefficient of Uniformity (Cu) used for?",
        "options": ["To determine the average particle size", "To classify the soil gradation", "To measure the particle shape", "To calculate the particle density"],
        "answer": "To classify the soil gradation"
    },
    {
        "question": "What does a high percentage of fines indicate about the soil?",
        "options": ["The soil is well-graded", "The soil is poorly graded", "The soil is fine-grained", "The soil is coarse-grained"],
        "answer": "The soil is fine-grained"
    }
]

# Function to create quiz
def create_quiz(questions):
    quiz_widgets = []
    for q in questions:
        question_widget = widgets.VBox([
            widgets.HTML(value=f"<b>{q['question']}</b>"),
            widgets.RadioButtons(options=q['options'], layout={'width': 'max-content'})
        ])
        quiz_widgets.append(question_widget)
    return quiz_widgets

# Function to check answers
def check_answers(quiz_widgets, questions):
    score = 0
    for i, widget in enumerate(quiz_widgets):
        selected = widget.children[1].value
        if selected == questions[i]['answer']:
            score += 1
    return score

# Display quiz
quiz_widgets = create_quiz(questions)
display(widgets.VBox(quiz_widgets))

# Button to submit answers
submit_button = widgets.Button(description="Submit Answers")
output = widgets.Output()

def on_submit(b):
    with output:
        output.clear_output()
        score = check_answers(quiz_widgets, questions)
        display(Markdown(f"### Your Score: {score}/{len(questions)}"))

submit_button.on_click(on_submit)
display(submit_button, output)