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 on the surface of a shaft in torsion is in pure shear with τₓᵧ = −70 MPa (σₓ = σᵧ = 0). Find the normal stress σ_θ on a plane whose normal is rotated 25° counter-clockwise from the x-axis.

τ_xy = −70 MPaθ = 25°

σ_θ = MPa

2.Easy

A point on the surface of a shaft in torsion is in pure shear with τₓᵧ = −60 MPa (σₓ = σᵧ = 0). Find the normal stress σ_θ on a plane whose normal is rotated 45° clockwise from the x-axis.

τ_xy = −60 MPaθ = -45°

σ_θ = MPa

3.Medium

A point in a loaded part has σₓ = −80 MPa, σᵧ = 20 MPa and τₓᵧ = 20 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 = −80 MPaσ_y = 20 MPaτ_xy = 20 MPa

θ_p = °

4.Medium

A point in a loaded part has σₓ = −70 MPa, σᵧ = 10 MPa and τₓᵧ = 30 MPa (tension positive). Find the maximum in-plane shear stress τ_max.

σ_x = −70 MPaσ_y = 10 MPaτ_xy = 30 MPa

τ_max = MPa

5.Hard

A solid shaft of diameter 50 mm carries an axial tensile force of 150 kN together with a torque of 1500 N·m. At the surface of the shaft, find the largest principal stress σ₁.

d = 50 mmF = 150 kNT = 1500 N·m

σ₁ = MPa

6.Hard

A point in a thin plate is in plane stress: σₓ = 150 MPa, σᵧ = −70 MPa and τₓᵧ = −20 MPa (σ_z = 0). Find the von Mises equivalent stress σ_vM.

σ_x = 150 MPaσ_y = −70 MPaτ_xy = −20 MPa

σ_vM = MPa

Answer key

1. σ_θ = −53.62 MPa
2. σ_θ = 60 MPa
3. θ_p = 79.1 °
4. τ_max = 50 MPa
5. σ₁ = 110.3 MPa
6. σ_vM = 197.7 MPa

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