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​ = 0σy​ = −10τxy​ = 20MPa

A point in a loaded part has σₓ = 0 MPa, σᵧ = −10 MPa and τₓᵧ = 20 MPa (tension positive). Find the largest principal stress σ1.

σx = 0 MPaσy = −10 MPaτxy = 20 MPa
σ1 = MPa
2.Easy
σx​ = 60σy​ = −60τxy​ = 30MPa

A point in a loaded part has σₓ = 60 MPa, σᵧ = −60 MPa and τₓᵧ = 30 MPa (tension positive). Find the largest principal stress σ1.

σx = 60 MPaσy = −60 MPaτxy = 30 MPa
σ1 = MPa
3.Medium
σx​ = 60σy​ = 90τxy​ = −50MPa

A point in a loaded part has σₓ = 60 MPa, σᵧ = 90 MPa and τₓᵧ = −50 MPa (tension positive). Find the maximum in-plane shear stress τmax.

σx = 60 MPaσy = 90 MPaτxy = −50 MPa
τmax = MPa
4.Medium
σx​ = 130σy​ = 80τxy​ = 20MPa

A point in a thin plate is in plane stress: σₓ = 130 MPa, σᵧ = 80 MPa and τₓᵧ = 20 MPa, with σz = 0. Find the absolute maximum shear stress τabs at the point (considering all planes in three dimensions).

σx = 130 MPaσy = 80 MPaτxy = 20 MPaσz = 0
τabs = MPa
5.Hard
σx​ = 10σy​ = 120τxy​ = −60MPa

A point in a thin plate is in plane stress: σₓ = 10 MPa, σᵧ = 120 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 von Mises (distortion energy) criterion.

σx = 10 MPaσy = 120 MPaτxy = −60 MPaσY = 420 MPa
n =
6.Hard
σx​τxy​top surface of the shaft

A solid shaft of diameter 30 mm carries a bending moment of 100 N·m together with a torque of 250 N·m. At the top surface, where the bending stress is tensile and largest, find the maximum in-plane shear stress τmax.

d = 30 mmM = 100 N·mT = 250 N·m
τmax = MPa

Answer key

  1. 1. σ1 = 15.62 MPa
  2. 2. σ1 = 67.08 MPa
  3. 3. τmax = 52.2 MPa
  4. 4. τabs = 68.51 MPa
  5. 5. n = 2.705
  6. 6. τmax = 50.79 MPa

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