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
σx​ = 50σy​ = −80τxy​MPa

A point has σₓ = 50 MPa and σᵧ = −80 MPa, plus some shear stress. On an element rotated by 60° the normal stress in the rotated x′ direction is −12.9 MPa. Find the normal stress σy′ in the rotated y′ direction.

σx = 50 MPaσy = −80 MPaσx′ = −12.9 MPaθ = 60°
σy′ = MPa
2.Easy
σx​ = 160σy​ = −20τxy​ = 40MPa

A point in a loaded part has σₓ = 160 MPa, σᵧ = −20 MPa and τₓᵧ = 40 MPa (tension positive). Find the smallest principal stress σ2.

σx = 160 MPaσy = −20 MPaτxy = 40 MPa
σ2 = MPa
3.Medium
σx​ = 60σy​ = −80τxy​ = 10MPa

A point in a loaded part has σₓ = 60 MPa, σᵧ = −80 MPa and τₓᵧ = 10 MPa (tension positive). Find the maximum in-plane shear stress τmax.

σx = 60 MPaσy = −80 MPaτxy = 10 MPa
τmax = MPa
4.Medium
σx​ = 40σy​ = −70τxy​ = 50MPa

A point has σₓ = 40 MPa, σᵧ = −70 MPa and τₓᵧ = 50 MPa (tension positive). The element is rotated 40° counter-clockwise. Find the normal stress σy′ on the faces of the rotated element that are perpendicular to the rotated y′-axis.

σx = 40 MPaσy = −70 MPaτxy = 50 MPaθ = 40°
σy′ = MPa
5.Hard
σx​ = 70σy​ = −60τxy​ = 50MPa

A point in a thin plate is in plane stress: σₓ = 70 MPa, σᵧ = −60 MPa and τₓᵧ = 50 MPa (σz = 0). The material has a yield strength of 300 MPa. Find the factor of safety n against yielding by the Tresca (maximum shear stress) criterion.

σx = 70 MPaσy = −60 MPaτxy = 50 MPaσY = 300 MPa
n =
6.Hard
σx​ = 150σy​ = −30τxy​ = −20MPa

A point in a loaded part has σₓ = 150 MPa, σᵧ = −30 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 = 150 MPaσy = −30 MPaτxy = −20 MPaθ = -20°
σθ = MPa

Answer key

  1. 1. σy′ = −17.1 MPa
  2. 2. σ2 = −28.49 MPa
  3. 3. τmax = 70.71 MPa
  4. 4. σy′ = −73.79 MPa
  5. 5. n = 1.829
  6. 6. σθ = 141.8 MPa

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