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 = 110 MPa, σᵧ = −50 MPa and τₓᵧ = 10 MPa (tension positive). Find the smallest principal stress σ₂.

σ_x = 110 MPaσ_y = −50 MPaτ_xy = 10 MPa

σ₂ = MPa

2.Easy

A bar with cross-sectional area 200 mm² carries an axial tensile force of 70 kN. Find the magnitude of the shear stress |τ_θ| on a section whose normal makes an angle of 35° with the bar axis.

P = 70 kNA = 200 mm²θ = 35°

|τ_θ| = MPa

3.Medium

A point in a loaded part has σₓ = 0 MPa, σᵧ = −30 MPa and τₓᵧ = 50 MPa (tension positive). Find the angle θ_p from the x-axis to the direction of the largest principal stress, between −90° and 90°.

σ_x = 0 MPaσ_y = −30 MPaτ_xy = 50 MPa

θ_p = °

4.Medium

A point has σₓ = 0 MPa and σᵧ = −60 MPa. How large can the shear stress τ_xy (magnitude) become before the largest principal stress reaches 15 MPa?

σ_x = 0 MPaσ_y = −60 MPaσ₁ = 15 MPa

τ_xy = MPa

5.Hard

A point in a thin plate is in plane stress: σₓ = 90 MPa, σᵧ = 110 MPa and τₓᵧ = 20 MPa (σ_z = 0). Find the von Mises equivalent stress σ_vM.

σ_x = 90 MPaσ_y = 110 MPaτ_xy = 20 MPa

σ_vM = MPa

6.Hard

A solid shaft of diameter 20 mm carries an axial compressive force of 120 kN together with a torque of 400 N·m. At the surface of the shaft, find the largest principal stress σ₁.

d = 20 mmF = −120 kNT = 400 N·m

σ₁ = MPa

Answer key

1. σ₂ = −50.62 MPa
2. |τ_θ| = 164.4 MPa
3. θ_p = 36.65 °
4. τ_xy = 33.54 MPa
5. σ_vM = 107.2 MPa
6. σ₁ = 127.3 MPa

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