Understanding the Mohr’s Circle formula is one of the most important concepts in Strength of Materials and mechanical engineering. It is widely used to determine principal stresses, maximum shear stress, and stress transformation in materials.
Many students face difficulty in solving Mohr’s Circle problems because they involve multiple steps and calculations. This is where a Mohr’s Circle calculator approach becomes extremely useful.
In real engineering applications like pressure vessels, shafts, and structural components, stress transformation is critical. A small mistake in calculation can lead to incorrect design and potential failure.
In this practical guide, you will learn:
- ✔ Complete Mohr’s Circle formula
- ✔ Step-by-step calculator usage
- ✔ Basic to advanced problems
1. What is Mohr’s Circle?
Mohr’s Circle is a graphical and mathematical method used to determine:
- Principal stresses
- Maximum shear stress
- Orientation of stress planes
It simplifies complex stress calculations into an easy-to-understand format.
2. Basic Mohr’s Circle Formula
Formula for Principal Stress:
σ₁, σ₂ = (σx + σy)/2 ± √[((σx − σy)/2)² + τxy²]
Meaning
- σ₁ = Maximum principal stress
- σ₂ = Minimum principal stress
- τxy = Shear stress
3. Calculator Example (Basic)
Given:
- σx = 100 MPa
- σy = 40 MPa
- τxy = 30 MPa
Step-by-Step Calculation
Step 1: (100 + 40) ÷ 2 = 70
Step 2: (100 − 40) ÷ 2 = 30
Step 3: 30² = 900
Step 4: 900 + 900 = 1800
Step 5: √1800 ≈ 42.43
✅ Final Answer
- σ₁ = 112.43 MPa
- σ₂ = 27.57 MPa
This is the most common and student-friendly method to solve the mohr circle formula.
4. Direct Calculator Method (Best)
👉 Instead of step-by-step, type directly:
((100 + 40) ÷ 2) + √(((100 − 40) ÷ 2)² + 30²)
Use this method for faster calculations once your concept is clear.
✔ Use √ and x² buttons carefully
5. Maximum Shear Stress Formula
Formula:
τmax = √[((σx − σy)/2)² + τxy²]
Example
τmax = √(900 + 900) = √1800 = 42.43 MPa
6. Angle of Principal Plane
Formula:
tan(2θ) = 2τxy / (σx − σy)
Example
(2 × 30) ÷ (100 − 40) = 60 ÷ 60 = 1
Calculator: tan⁻¹(1) = 45°
👉 θ = 45° ÷ 2 = 22.5°
7. Advanced Calculator Problem
Given:
- σx = 120
- σy = 60
- τxy = 40
Calculator Input:
((120 + 60)/2) + √(((120 − 60)/2)² + 40²)
Steps
- Average = 90
- Difference = 30
- 30² = 900
- 40² = 1600
- Total = 2500
- √2500 = 50
✅ Final Answer
- σ₁ = 140 MPa
- σ₂ = 40 MPa
8. Scientific Calculator Tips
- ✔ Use brackets ( ) properly
- ✔ Use square (x²) and root (√)
- ✔ Solve step-by-step if confused
- ✔ Use tan⁻¹ for angle
Common Mistakes
- ❌ Wrong sign (+ / − confusion)
- ❌ Missing square root
- ❌ Calculator bracket error
- ❌ Angle calculation mistake
Practice Questions
- ((80 + 20)/2) ± √(((80 − 20)/2)² + 25²)
- ((150 + 50)/2) ± √(((150 − 50)/2)² + 40²)
✅ Answers
Solve step-by-step using calculator
Conclusion
Mastering the Mohr’s Circle formula is essential for solving real engineering stress problems.
If you understand:
- ✔ Formula
- ✔ Calculator usage
- ✔ Step-by-step solving
Start practicing today using your online scientific calculator .
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