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​ = 50MPapure shear

A point on the surface of a shaft in torsion is in pure shear with τₓᵧ = 50 MPa (σₓ = σᵧ = 0). Find the normal stress σθ on a plane whose normal is rotated 30° counter-clockwise from the x-axis.

τxy = 50 MPaθ = 30°
σθ = MPa
2.Easy
σx​ = 40σy​ = 100τxy​ = 40MPa

A point in a loaded part has σₓ = 40 MPa, σᵧ = 100 MPa and τₓᵧ = 40 MPa (tension positive). Find the largest principal stress σ1.

σx = 40 MPaσy = 100 MPaτxy = 40 MPa
σ1 = MPa
3.Medium
σx​ = −30σy​ = −50τxy​ = −60MPa

A point in a loaded part has σₓ = −30 MPa, σᵧ = −50 MPa and τₓᵧ = −60 MPa (tension positive). Find the maximum in-plane shear stress τmax.

σx = −30 MPaσy = −50 MPaτxy = −60 MPa
τmax = MPa
4.Medium
σx​ = −30σy​ = −80τxy​ = 30MPa

A point in a thin plate is in plane stress: σₓ = −30 MPa, σᵧ = −80 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 = −30 MPaσy = −80 MPaτxy = 30 MPaσz = 0
τabs = MPa
5.Hard
σx​ = 80σy​ = −30τxy​ = −10MPa

A point in a loaded part has σₓ = 80 MPa, σᵧ = −30 MPa and τₓᵧ = −10 MPa (tension positive). Find the normal stress σθ on a plane whose normal is rotated 60° counter-clockwise from the x-axis.

σx = 80 MPaσy = −30 MPaτxy = −10 MPaθ = 60°
σθ = MPa
6.Hard
σx​ = 40σy​ = 10τxy​ = −40MPa

A point in a thin plate is in plane stress: σₓ = 40 MPa, σᵧ = 10 MPa and τₓᵧ = −40 MPa (σz = 0). The material has a yield strength of 275 MPa. Find the factor of safety n against yielding by the von Mises (distortion energy) criterion.

σx = 40 MPaσy = 10 MPaτxy = −40 MPaσY = 275 MPa
n =

Answer key

  1. 1. σθ = 43.3 MPa
  2. 2. σ1 = 120 MPa
  3. 3. τmax = 60.83 MPa
  4. 4. τabs = 47.03 MPa
  5. 5. σθ = −11.16 MPa
  6. 6. n = 3.521

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