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 = −20 MPa (tension positive). Find the smallest principal stress σ₂.

σ_x = 110 MPaσ_y = 50 MPaτ_xy = −20 MPa

σ₂ = MPa

2.Easy

A point has σₓ = −20 MPa and σᵧ = 100 MPa, plus some shear stress. On an element rotated by 60° the normal stress in the rotated x′ direction is 113.3 MPa. Find the normal stress σ_y′ in the rotated y′ direction.

σ_x = −20 MPaσ_y = 100 MPaσ_x′ = 113.3 MPaθ = 60°

σ_y′ = MPa

3.Medium

A point in a loaded part has σₓ = 110 MPa, σᵧ = 0 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 = 110 MPaσ_y = 0 MPaτ_xy = 20 MPa

θ_p = °

4.Medium

A point in a loaded part has σₓ = 130 MPa, σᵧ = 10 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 = 130 MPaσ_y = 10 MPaτ_xy = −20 MPa

θ_p = °

5.Hard

A point in a loaded part has σₓ = −50 MPa, σᵧ = 90 MPa and τₓᵧ = 10 MPa (tension positive). Find the normal stress σ_θ on a plane whose normal is rotated 50° counter-clockwise from the x-axis.

σ_x = −50 MPaσ_y = 90 MPaτ_xy = 10 MPaθ = 50°

σ_θ = MPa

6.Hard

A solid shaft of diameter 40 mm carries a bending moment of 300 N·m together with a torque of 2000 N·m. At the top surface, where the bending stress is tensile and largest, find the largest principal stress σ₁.

d = 40 mmM = 300 N·mT = 2000 N·m

σ₁ = MPa

Answer key

1. σ₂ = 43.94 MPa
2. σ_y′ = −33.3 MPa
3. θ_p = 9.992 °
4. θ_p = −9.217 °
5. σ_θ = 42 MPa
6. σ₁ = 184.8 MPa

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