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​ = −30τxy​ = −20MPa

A point in a loaded part has σₓ = 50 MPa, σᵧ = −30 MPa and τₓᵧ = −20 MPa (tension positive). Find the smallest principal stress σ2.

σx = 50 MPaσy = −30 MPaτxy = −20 MPa
σ2 = MPa
2.Easy
σx​ = 200σy​ = −110MPaprincipal element

At a point in a loaded part the in-plane principal stresses are σ₁ = 200 MPa and σ₂ = −110 MPa (tension positive). Find the normal stress σavg that acts on the planes of maximum in-plane shear stress.

σ1 = 200 MPaσ2 = −110 MPa
σavg = MPa
3.Medium
σx​ = −10σy​ = 10MPaτxy​ = ?

A point has σₓ = −10 MPa and σᵧ = 10 MPa. How large can the shear stress τxy (magnitude) become before the largest principal stress reaches 50 MPa?

σx = −10 MPaσy = 10 MPaσ1 = 50 MPa
τxy = MPa
4.Medium
σx​ = 150σy​ = 60τxy​ = −50MPa

A point in a thin plate is in plane stress: σₓ = 150 MPa, σᵧ = 60 MPa and τₓᵧ = −50 MPa, with σz = 0. Find the absolute maximum shear stress τabs at the point (considering all planes in three dimensions).

σx = 150 MPaσy = 60 MPaτxy = −50 MPaσz = 0
τabs = MPa
5.Hard
σx​ = 130σy​ = −30τxy​ = −20MPa

A point in a thin plate is in plane stress: σₓ = 130 MPa, σᵧ = −30 MPa and τₓᵧ = −20 MPa (σz = 0). Find the von Mises equivalent stress σvM.

σx = 130 MPaσy = −30 MPaτxy = −20 MPa
σvM = MPa
6.Hard
σx​ = 100σy​ = 110τxy​ = −60MPa

A point in a thin plate is in plane stress: σₓ = 100 MPa, σᵧ = 110 MPa and τₓᵧ = −60 MPa (σz = 0). The material has a yield strength of 420 MPa. Find the factor of safety n against yielding by the Tresca (maximum shear stress) criterion.

σx = 100 MPaσy = 110 MPaτxy = −60 MPaσY = 420 MPa
n =

Answer key

  1. 1. σ2 = −34.72 MPa
  2. 2. σavg = 45 MPa
  3. 3. τxy = 48.99 MPa
  4. 4. τabs = 86.13 MPa
  5. 5. σvM = 151.3 MPa
  6. 6. n = 2.542

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