Mohr's Circle

Why it matters

Mohr's Circle is a crucial tool in mechanical engineering for visualizing and analyzing the state of stress at a point in a material. It helps engineers determine principal stresses and maximum shear stresses, which are essential for ensuring the safety and reliability of structures and mechanical components.

Key ideas

• Stress Transformation: Mohr's Circle provides a graphical representation of the transformation of stresses in a material subjected to complex loading.
• Principal Stresses: These are the normal stresses acting on a plane where shear stress is zero. Mohr's Circle helps in identifying these stresses.
• Maximum Shear Stress: The circle also helps in determining the maximum shear stress, which is critical for failure analysis.
• Graphical Representation: By plotting normal stress (σ) on the x-axis and shear stress (τ) on the y-axis, Mohr's Circle provides a visual method to solve stress transformation equations.

Construction and scope

For the in-plane stress circle, plot A = (σ_x, τ_xy) and B = (σ_y, −τ_xy) as opposite ends of a diameter, using one consistent shear-sign convention. The centre is C = ((σ_x + σ_y)/2, 0). The radius gives the in-plane maximum shear. A physical plane rotation θ corresponds to twice that angle on the circle; the plotted direction depends on the chosen shear-axis convention.

For plane stress, include the third principal stress 0 when computing the absolute maximum shear. In the example, the three principal stresses are 110.36, 39.64, and 0 MPa, so the absolute maximum shear is (110.36 − 0)/2 = 55.18 MPa. The two-dimensional circle radius remains 35.36 MPa.

Formulas

• σ₁, σ₂ = (σₓ + σᵧ)/2 ± √[((σₓ - σᵧ)/2)² + τₓᵧ²]
  • σ₁, σ₂: Principal stresses (Pa)
  • σₓ, σᵧ: Normal stresses on x and y planes (Pa)
  • τₓᵧ: Shear stress on the xy plane (Pa)
• τ_max = √[((σₓ - σᵧ)/2)² + τₓᵧ²]
  • τ_max: Maximum shear stress (Pa)

Worked example

Given:

• Normal stress on x-plane, σₓ = 100 MPa
• Normal stress on y-plane, σᵧ = 50 MPa
• Shear stress on xy-plane, τₓᵧ = 25 MPa

Steps:

1. Calculate the average normal stress:
   • σ_avg = (σₓ + σᵧ) / 2 = (100 + 50) / 2 = 75 MPa
2. Calculate the radius of Mohr's Circle:
   • R = √[((σₓ - σᵧ)/2)² + τₓᵧ²] = √[((100 - 50)/2)² + 25²] = √[625 + 625] = √1250 = 35.36 MPa
3. Determine the principal stresses:
   • σ₁ = σ_avg + R = 75 + 35.36 = 110.36 MPa
   • σ₂ = σ_avg - R = 75 - 35.36 = 39.64 MPa
4. Calculate the maximum shear stress:
   • τ_max = R = 35.36 MPa

Final Answer:

• Principal stresses: σ₁ = 110.36 MPa, σ₂ = 39.64 MPa
• In-plane maximum shear stress: τ_max = 35.36 MPa

Common mistakes

• Confusing the axes: Remember that normal stress is plotted on the x-axis and shear stress on the y-axis.
• Incorrect calculation of the radius of Mohr's Circle, which affects the determination of principal stresses.
• Forgetting to convert units to SI before calculations.

For GATE ME

Questions on Mohr's Circle often involve calculating principal stresses, maximum shear stresses, and stress transformation. Practice problems that require drawing Mohr's Circle and interpreting it for different stress states.

Quick check

1. What does Mohr's Circle help determine in a material?
2. On which axis is shear stress plotted in Mohr's Circle?
3. How do you calculate the radius of Mohr's Circle?

Answers: 1. Principal and maximum shear stresses, 2. y-axis, 3. R = √[((σₓ - σᵧ)/2)² + τₓᵧ²]