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 = 10 MPa, σᵧ = 0 MPa and τₓᵧ = 60 MPa (tension positive). Find the largest principal stress σ₁.

σ_x = 10 MPaσ_y = 0 MPaτ_xy = 60 MPa

σ₁ = MPa

2.Easy

A point has σₓ = 120 MPa and σᵧ = 10 MPa, plus some shear stress. On an element rotated by 15° the normal stress in the rotated x′ direction is 122.6 MPa. Find the normal stress σ_y′ in the rotated y′ direction.

σ_x = 120 MPaσ_y = 10 MPaσ_x′ = 122.6 MPaθ = 15°

σ_y′ = MPa

3.Medium

A point has σₓ = 50 MPa and σᵧ = −20 MPa. How large can the shear stress τ_xy (magnitude) become before the largest principal stress reaches 60 MPa?

σ_x = 50 MPaσ_y = −20 MPaσ₁ = 60 MPa

τ_xy = MPa

4.Medium

A point in a loaded part has σₓ = 30 MPa, σᵧ = −60 MPa and τₓᵧ = 40 MPa (tension positive). Find the maximum in-plane shear stress τ_max.

σ_x = 30 MPaσ_y = −60 MPaτ_xy = 40 MPa

τ_max = MPa

5.Hard

A solid shaft of diameter 25 mm carries an axial compressive force of 70 kN together with a torque of 750 N·m. At the surface of the shaft, find the largest principal stress σ₁.

d = 25 mmF = −70 kNT = 750 N·m

σ₁ = MPa

6.Hard

A solid shaft of diameter 30 mm carries a bending moment of 1000 N·m together with a torque of 200 N·m. At the top surface, where the bending stress is tensile and largest, find the largest principal stress σ₁.

d = 30 mmM = 1000 N·mT = 200 N·m

σ₁ = MPa

Answer key

1. σ₁ = 65.21 MPa
2. σ_y′ = 7.4 MPa
3. τ_xy = 28.28 MPa
4. τ_max = 60.21 MPa
5. σ₁ = 183.3 MPa
6. σ₁ = 381 MPa

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