Two‑Mohr Circle Triaxial Calculator — φ and c from two tests

Two‑Mohr Circle Triaxial Calculator

Compute φ (deg), cohesion c (kPa), draw two Mohr circles & failure envelope, predict σ₁ for new σ₃

Input — Two consolidated drained triaxial tests

Test 1: σ₃ (kPa)

Test 1: σ₁ (kPa)

Test 2: σ₃ (kPa)

Test 2: σ₁ (kPa)

Predict σ₁ for new σ₃

σ₃ (kPa) to predict

 

Computation steps & formulas

We use the linearised triaxial failure relation: σ₁ = σ₃ · tan²(45° + φ/2) + 2c · tan(45° + φ/2)
Set Y = σ₁ and X = σ₃, then Y = mX + b where m = tan²(45° + φ/2) and b = 2c · tan(45° + φ/2). From two tests compute slope m, then extract φ and c.

Mohr diagram — σ (horizontal) vs τ (vertical)

Notes: Mohr circles are plotted for the two tests using centers (σ_avg,0) and radius (σ₁−σ₃)/2. The failure envelope τ = σ·tan(φ) + c is shown (in σ–τ space). Use different inputs to update the plot and results.

Detailed Step‑by‑Step Results

🧭 Mohr’s Circle & Shear Strength Calculator – User Guide

📌 Introduction

This tool is used to determine soil shear strength parameters using results from two triaxial tests. It calculates:

• Angle of internal friction (φ)
• Cohesion (c)
• Failure envelope
• Predicted major principal stress (σ₁)
• Mohr’s Circle diagram

⚙️ Step 1: Enter Test Data

You must enter values from two triaxial tests:

🔢 Required Inputs

• σ₃₁ – Minor principal stress for Test 1 (kPa)
• σ₁₁ – Major principal stress for Test 1 (kPa)
• σ₃₂ – Minor principal stress for Test 2 (kPa)
• σ₁₂ – Major principal stress for Test 2 (kPa)

➕ Optional Input

• σ₃ (new) – New confining stress to predict σ₁

💡 Tip: Use laboratory triaxial test results for accurate calculations.

▶️ Step 2: Run the Calculation

Click the Compute button or press Enter. The tool automatically performs:

• Linear regression between test points
• Determination of slope (m)
• Calculation of friction angle (φ)
• Calculation of cohesion (c)
• Prediction of σ₁ for new σ₃

📊 Step 3: Understand the Results

1. Slope (m)

Represents relationship:

σ₁ = mσ₃ + B

2. Angle of Internal Friction (φ)

• Calculated from slope
• Represents shear resistance of soil

3. Cohesion (c)

• Intercept of failure envelope
• Represents bonding between soil particles

4. Predicted σ₁

• Computed using entered σ₃ (new)
• Useful for design analysis

📐 Step 4: Mohr’s Circle Diagram

The tool automatically plots:

• Two Mohr’s circles (from test data)
• Failure envelope line
• Principal stresses (σ₁ and σ₃)

📌 Interpretation

• Circle touching envelope → Failure condition
• Envelope slope → tan(φ)
• Intercept → cohesion (c)

📘 Step 5: Review Calculation Steps

The tool displays:

• Formula substitutions
• Intermediate values
• Verification checks

This helps in understanding the calculation process and verifying results.

⚠️ Input Validation Rules

• All values must be positive
• σ₁ must be greater than σ₃
• σ₃ values must be different for both tests
• Slope must be positive
• Friction angle must be between 0° and 90°

⚠ Invalid inputs will display an error message and stop calculation.

📈 Engineering Significance

• φ (Friction Angle) → Governs shear strength in granular soils
• c (Cohesion) → Important for clayey soils
• Mohr Circle → Visual representation of stress state
• Failure Envelope → Defines shear failure condition

💡 Practical Tips

• Use accurate lab data for best results
• Avoid very small differences in σ₃ values
• Check if cohesion is negative (may indicate data issue)
• Use multiple tests for better reliability

🚀 Summary

This tool simplifies soil mechanics calculations by:

• Automating Mohr’s Circle construction
• Calculating φ and c instantly
• Providing visual and numerical outputs
• Helping in geotechnical design decisions