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
τxy​ = 120MPapure shear

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 45° clockwise from the x-axis.

τxy = 120 MPaθ = -45°
σθ = MPa
2.Easy
σx​ = −50σy​ = 30τxy​MPa

A point has σₓ = −50 MPa and σᵧ = 30 MPa, plus some shear stress. On an element rotated by 30° the normal stress in the rotated x′ direction is −73.3 MPa. Find the normal stress σy′ in the rotated y′ direction.

σx = −50 MPaσy = 30 MPaσx′ = −73.3 MPaθ = 30°
σy′ = MPa
3.Medium
σx​ = 160σy​ = 90τxy​ = −30MPa

A point in a thin plate is in plane stress: σₓ = 160 MPa, σᵧ = 90 MPa and τₓᵧ = −30 MPa, with σz = 0. Find the absolute maximum shear stress τabs at the point (considering all planes in three dimensions).

σx = 160 MPaσy = 90 MPaτxy = −30 MPaσz = 0
τabs = MPa
4.Medium
σx​ = 30σy​ = −70τxy​ = −60MPa

A point in a loaded part has σₓ = 30 MPa, σᵧ = −70 MPa and τₓᵧ = −60 MPa (tension positive). Find the maximum in-plane shear stress τmax.

σx = 30 MPaσy = −70 MPaτxy = −60 MPa
τmax = MPa
5.Hard
σx​ = −50σy​ = 10τxy​ = 20MPa

A point in a loaded part has σₓ = −50 MPa, σᵧ = 10 MPa and τₓᵧ = 20 MPa (tension positive). Find the normal stress σθ on a plane whose normal is rotated 20° clockwise from the x-axis.

σx = −50 MPaσy = 10 MPaτxy = 20 MPaθ = -20°
σθ = MPa
6.Hard
σx​ = −60σy​ = −70τxy​ = −50MPa

A point in a thin plate is in plane stress: σₓ = −60 MPa, σᵧ = −70 MPa and τₓᵧ = −50 MPa (σz = 0). The material has a yield strength of 350 MPa. Find the factor of safety n against yielding by the von Mises (distortion energy) criterion.

σx = −60 MPaσy = −70 MPaτxy = −50 MPaσY = 350 MPa
n =

Answer key

  1. 1. σθ = −120 MPa
  2. 2. σy′ = 53.3 MPa
  3. 3. τabs = 85.55 MPa
  4. 4. τmax = 78.1 MPa
  5. 5. σθ = −55.84 MPa
  6. 6. n = 3.222

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