Mohr’s circle & stress transformation worksheet with answers

Free printable Mohr’s circle & stress transformation worksheet: 6 problems from easy to exam level, with an answer key. Print it or save it as a PDF.

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1.Easy

A point in a loaded part has σₓ = 150 MPa, σᵧ = −20 MPa and τₓᵧ = 30 MPa (tension positive). Find the largest principal stress σ₁.

σ_x = 150 MPaσ_y = −20 MPaτ_xy = 30 MPa

σ₁ = MPa

2.Easy

A bar with cross-sectional area 1000 mm² carries an axial compressive force of 160 kN. Find the normal stress σ_θ on a section whose normal makes an angle of 25° with the bar axis (tension positive).

P = −160 kNA = 1000 mm²θ = 25°

σ_θ = MPa

3.Medium

A point in a loaded part has σₓ = −40 MPa, σᵧ = 70 MPa and τₓᵧ = −40 MPa (tension positive). Find the maximum in-plane shear stress τ_max.

σ_x = −40 MPaσ_y = 70 MPaτ_xy = −40 MPa

τ_max = MPa

4.Medium

A point has σₓ = 120 MPa and σᵧ = −20 MPa. How large can the shear stress τ_xy (magnitude) become before the largest principal stress reaches 160 MPa?

σ_x = 120 MPaσ_y = −20 MPaσ₁ = 160 MPa

τ_xy = MPa

5.Hard

A point in a loaded part has σₓ = 60 MPa, σᵧ = 90 MPa and τₓᵧ = 20 MPa (tension positive). Find the shear stress τ_θ on a plane whose normal is rotated 60° counter-clockwise from the x-axis (positive in the same sense as τₓᵧ).

σ_x = 60 MPaσ_y = 90 MPaτ_xy = 20 MPaθ = 60°

τ_θ = MPa

6.Hard

A solid shaft of diameter 20 mm carries an axial tensile force of 50 kN together with a torque of 200 N·m. At the surface of the shaft, find the largest principal stress σ₁.

d = 20 mmF = 50 kNT = 200 N·m

σ₁ = MPa

Answer key

1. σ₁ = 155.1 MPa
2. σ_θ = −131.4 MPa
3. τ_max = 68.01 MPa
4. τ_xy = 84.85 MPa
5. τ_θ = 2.99 MPa
6. σ₁ = 229.7 MPa

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