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 35° counter-clockwise from the x-axis.

τ_xy = −70 MPaθ = 35°

σ_θ = MPa

2.Easy

A point in a loaded part has σₓ = 10 MPa, σᵧ = −10 MPa and τₓᵧ = 50 MPa (tension positive). Find the largest principal stress σ₁.

σ_x = 10 MPaσ_y = −10 MPaτ_xy = 50 MPa

σ₁ = MPa

3.Medium

A point has σₓ = 30 MPa, σᵧ = 10 MPa and τₓᵧ = −30 MPa (tension positive). Find the angle θ_s, between −90° and 90°, from the x-axis to the normal of the plane on which the shear stress has its maximum positive value +R (shear positive in the same sense as τₓᵧ).

σ_x = 30 MPaσ_y = 10 MPaτ_xy = −30 MPa

θ_s = °

4.Medium

A point has σₓ = 30 MPa and σᵧ = 100 MPa. How large can the shear stress τ_xy (magnitude) become before the largest principal stress reaches 120 MPa?

σ_x = 30 MPaσ_y = 100 MPaσ₁ = 120 MPa

τ_xy = MPa

5.Hard

A point in a loaded part has σₓ = −70 MPa, σᵧ = 80 MPa and τₓᵧ = 30 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 = −70 MPaσ_y = 80 MPaτ_xy = 30 MPaθ = 50°

σ_θ = MPa

6.Hard

A point in a loaded part has σₓ = 90 MPa, σᵧ = 100 MPa and τₓᵧ = −50 MPa (tension positive). Find the normal stress σ_θ on a plane whose normal is rotated 45° counter-clockwise from the x-axis.

σ_x = 90 MPaσ_y = 100 MPaτ_xy = −50 MPaθ = 45°

σ_θ = MPa

Answer key

1. σ_θ = −65.78 MPa
2. σ₁ = 50.99 MPa
3. θ_s = −80.78 °
4. τ_xy = 42.43 MPa
5. σ_θ = 47.57 MPa
6. σ_θ = 45 MPa

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