1.Easy
A point in a loaded part has σₓ = 90 MPa, σᵧ = 20 MPa and τₓᵧ = 30 MPa (tension positive). Find the smallest principal stress σ2.
σx = 90 MPaσy = 20 MPaτxy = 30 MPa
σ2 = MPa
Mohr’s circle & stress transformation worksheet with answers
2.Easy
A point on the surface of a shaft in torsion is in pure shear with τₓᵧ = 120 MPa (σₓ = σᵧ = 0). Find the normal stress σθ on a plane whose normal is rotated 20° counter-clockwise from the x-axis.
τxy = 120 MPaθ = 20°
σθ = MPa
3.Medium
A point has σₓ = −10 MPa and σᵧ = 60 MPa. How large can the shear stress τxy (magnitude) become before the largest principal stress reaches 100 MPa?
σx = −10 MPaσy = 60 MPaσ1 = 100 MPa
τxy = MPa
4.Medium
A point has σₓ = 150 MPa, σᵧ = −30 MPa and τₓᵧ = 10 MPa (tension positive). The element is rotated 35° counter-clockwise. Find the normal stress σy′ on the faces of the rotated element that are perpendicular to the rotated y′-axis.
σx = 150 MPaσy = −30 MPaτxy = 10 MPaθ = 35°
σy′ = MPa
5.Hard
A point in a loaded part has σₓ = 50 MPa, σᵧ = −70 MPa and τₓᵧ = −40 MPa (tension positive). Find the normal stress σθ on a plane whose normal is rotated 45° counter-clockwise from the x-axis.
σx = 50 MPaσy = −70 MPaτxy = −40 MPaθ = 45°
σθ = MPa
6.Hard
A point in a loaded part has σₓ = 20 MPa, σᵧ = 40 MPa and τₓᵧ = 30 MPa (tension positive). Find the shear stress τθ on a plane whose normal is rotated 20° counter-clockwise from the x-axis (positive in the same sense as τₓᵧ).
σx = 20 MPaσy = 40 MPaτxy = 30 MPaθ = 20°
τθ = MPa
Answer key
- 1. σ2 = 8.902 MPa
- 2. σθ = 77.13 MPa
- 3. τxy = 66.33 MPa
- 4. σy′ = 19.82 MPa
- 5. σθ = −50 MPa
- 6. τθ = 29.41 MPa
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