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​ = −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 30° counter-clockwise from the x-axis.

τxy = −110 MPaθ = 30°
σθ = MPa
2.Easy
σx​ = 170σy​ = −70MPaprincipal element

At a point in a loaded part the in-plane principal stresses are σ₁ = 170 MPa and σ₂ = −70 MPa (tension positive). Find the maximum in-plane shear stress τmax.

σ1 = 170 MPaσ2 = −70 MPa
τmax = MPa
3.Medium
σx​ = 70σy​ = 70τxy​ = 60MPa

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

σx = 70 MPaσy = 70 MPaτxy = 60 MPa
τmax = MPa
4.Medium
σx​ = −20σy​ = 120τxy​ = −60MPa

A point in a loaded part has σₓ = −20 MPa, σᵧ = 120 MPa and τₓᵧ = −60 MPa (tension positive). Find the maximum in-plane shear stress τmax.

σx = −20 MPaσy = 120 MPaτxy = −60 MPa
τmax = MPa
5.Hard
σx​ = −60σy​ = 100τxy​ = −20MPa

A point in a thin plate is in plane stress: σₓ = −60 MPa, σᵧ = 100 MPa and τₓᵧ = −20 MPa (σz = 0). The material has a yield strength of 250 MPa. Find the factor of safety n against yielding by the Tresca (maximum shear stress) criterion.

σx = −60 MPaσy = 100 MPaτxy = −20 MPaσY = 250 MPa
n =
6.Hard
σx​ = 130σy​ = 40τxy​ = 10MPa

A point in a loaded part has σₓ = 130 MPa, σᵧ = 40 MPa and τₓᵧ = 10 MPa (tension positive). Find the shear stress τθ on a plane whose normal is rotated 30° counter-clockwise from the x-axis (positive in the same sense as τₓᵧ).

σx = 130 MPaσy = 40 MPaτxy = 10 MPaθ = 30°
τθ = MPa

Answer key

  1. 1. σθ = −95.26 MPa
  2. 2. τmax = 120 MPa
  3. 3. τmax = 60 MPa
  4. 4. τmax = 92.2 MPa
  5. 5. n = 1.516
  6. 6. τθ = −33.97 MPa

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