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​ = −70σy​ = −80τxy​ = −10MPa

A point in a loaded part has σₓ = −70 MPa, σᵧ = −80 MPa and τₓᵧ = −10 MPa (tension positive). Find the smallest principal stress σ2.

σx = −70 MPaσy = −80 MPaτxy = −10 MPa
σ2 = MPa
2.Easy
τxy​ = −110MPapure shear

A point on the surface of a shaft in torsion is in pure shear with τₓᵧ = −110 MPa (σₓ = σᵧ = 0). Find the normal stress σθ on a plane whose normal is rotated 35° counter-clockwise from the x-axis.

τxy = −110 MPaθ = 35°
σθ = MPa
3.Medium
σx​ = 160σy​ = −30τxy​ = −30MPa

A point in a loaded part has σₓ = 160 MPa, σᵧ = −30 MPa and τₓᵧ = −30 MPa (tension positive). Find the maximum in-plane shear stress τmax.

σx = 160 MPaσy = −30 MPaτxy = −30 MPa
τmax = MPa
4.Medium
σx​ = 10σy​ = 90τxy​ = 50MPa

A point has σₓ = 10 MPa, σᵧ = 90 MPa and τₓᵧ = 50 MPa (tension positive). The element is rotated 20° counter-clockwise. Find the normal stress σy′ on the faces of the rotated element that are perpendicular to the rotated y′-axis.

σx = 10 MPaσy = 90 MPaτxy = 50 MPaθ = 20°
σy′ = MPa
5.Hard
σx​ = 110σy​ = −40τxy​ = 10MPa

A point in a thin plate is in plane stress: σₓ = 110 MPa, σᵧ = −40 MPa and τₓᵧ = 10 MPa (σz = 0). Find the von Mises equivalent stress σvM.

σx = 110 MPaσy = −40 MPaτxy = 10 MPa
σvM = MPa
6.Hard
σx​ = 40σy​ = 110τxy​ = 10MPa

A point in a thin plate is in plane stress: σₓ = 40 MPa, σᵧ = 110 MPa and τₓᵧ = 10 MPa (σz = 0). The material has a yield strength of 355 MPa. Find the factor of safety n against yielding by the von Mises (distortion energy) criterion.

σx = 40 MPaσy = 110 MPaτxy = 10 MPaσY = 355 MPa
n =

Answer key

  1. 1. σ2 = −86.18 MPa
  2. 2. σθ = −103.4 MPa
  3. 3. τmax = 99.62 MPa
  4. 4. σy′ = 48.5 MPa
  5. 5. σvM = 135.6 MPa
  6. 6. n = 3.623

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