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 = 50 MPa, σᵧ = −50 MPa and τₓᵧ = 30 MPa (tension positive). Find the smallest principal stress σ₂.

σ_x = 50 MPaσ_y = −50 MPaτ_xy = 30 MPa

σ₂ = MPa

2.Easy

A point in a loaded part has σₓ = 120 MPa, σᵧ = 60 MPa and τₓᵧ = 20 MPa (tension positive). Find the largest principal stress σ₁.

σ_x = 120 MPaσ_y = 60 MPaτ_xy = 20 MPa

σ₁ = MPa

3.Medium

A point has σₓ = 100 MPa, σᵧ = 10 MPa and τₓᵧ = 60 MPa (tension positive). The element is rotated 15° counter-clockwise. Find the normal stress σ_y′ on the faces of the rotated element that are perpendicular to the rotated y′-axis.

σ_x = 100 MPaσ_y = 10 MPaτ_xy = 60 MPaθ = 15°

σ_y′ = MPa

4.Medium

A point in a loaded part has σₓ = −50 MPa, σᵧ = 120 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 = −50 MPaσ_y = 120 MPaτ_xy = −20 MPa

θ_p = °

5.Hard

A point in a loaded part has σₓ = 150 MPa, σᵧ = 70 MPa and τₓᵧ = −20 MPa (tension positive). Find the normal stress σ_θ on a plane whose normal is rotated 30° clockwise from the x-axis.

σ_x = 150 MPaσ_y = 70 MPaτ_xy = −20 MPaθ = -30°

σ_θ = MPa

6.Hard

A solid shaft of diameter 50 mm carries an axial compressive force of 80 kN together with a torque of 1000 N·m. At the surface of the shaft, find the largest principal stress σ₁.

d = 50 mmF = −80 kNT = 1000 N·m

σ₁ = MPa

Answer key

1. σ₂ = −58.31 MPa
2. σ₁ = 126.1 MPa
3. σ_y′ = −13.97 MPa
4. θ_p = −83.38 °
5. σ_θ = 147.3 MPa
6. σ₁ = 25.18 MPa

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