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​ = 120σy​ = −30τxy​ = 60MPa

A point in a loaded part has σₓ = 120 MPa, σᵧ = −30 MPa and τₓᵧ = 60 MPa (tension positive). Find the smallest principal stress σ2.

σx = 120 MPaσy = −30 MPaτxy = 60 MPa
σ2 = MPa
2.Easy
σx​ = −80σy​ = −40τxy​ = 10MPa

A point in a loaded part has σₓ = −80 MPa, σᵧ = −40 MPa and τₓᵧ = 10 MPa (tension positive). Find the largest principal stress σ1.

σx = −80 MPaσy = −40 MPaτxy = 10 MPa
σ1 = MPa
3.Medium
σx​ = 90σy​ = 100τxy​ = 30MPa

A point in a thin plate is in plane stress: σₓ = 90 MPa, σᵧ = 100 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 = 90 MPaσy = 100 MPaτxy = 30 MPaσz = 0
τabs = MPa
4.Medium
σx​ = 90σy​ = 120τxy​ = 40MPa

A point in a loaded part has σₓ = 90 MPa, σᵧ = 120 MPa and τₓᵧ = 40 MPa (tension positive). Find the angle θp from the x-axis to the direction of the largest principal stress, between −90° and 90°.

σx = 90 MPaσy = 120 MPaτxy = 40 MPa
θp = °
5.Hard
σx​ = 140σy​ = −10τxy​ = −40MPa

A point in a thin plate is in plane stress: σₓ = 140 MPa, σᵧ = −10 MPa and τₓᵧ = −40 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 = 140 MPaσy = −10 MPaτxy = −40 MPaσY = 300 MPa
n =
6.Hard
σx​ = 80σy​ = 60τxy​ = −20MPa

A point in a loaded part has σₓ = 80 MPa, σᵧ = 60 MPa and τₓᵧ = −20 MPa (tension positive). Find the shear stress τθ on a plane whose normal is rotated 20° clockwise from the x-axis (positive in the same sense as τₓᵧ).

σx = 80 MPaσy = 60 MPaτxy = −20 MPaθ = -20°
τθ = MPa

Answer key

  1. 1. σ2 = −51.05 MPa
  2. 2. σ1 = −37.64 MPa
  3. 3. τabs = 62.71 MPa
  4. 4. θp = 55.28 °
  5. 5. n = 1.765
  6. 6. τθ = −8.893 MPa

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