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​ = −60σy​ = 120τxy​MPa

A point has σₓ = −60 MPa and σᵧ = 120 MPa, plus some shear stress. On an element rotated by 20° the normal stress in the rotated x′ direction is −51.8 MPa. Find the normal stress σy′ in the rotated y′ direction.

σx = −60 MPaσy = 120 MPaσx′ = −51.8 MPaθ = 20°
σy′ = MPa
2.Easy
σx​ = 100σy​ = −20τxy​ = 30MPa

A point in a loaded part has σₓ = 100 MPa, σᵧ = −20 MPa and τₓᵧ = 30 MPa (tension positive). Find the largest principal stress σ1.

σx = 100 MPaσy = −20 MPaτxy = 30 MPa
σ1 = MPa
3.Medium
σx​ = −80σy​ = −70τxy​ = −10MPa

A point has σₓ = −80 MPa, σᵧ = −70 MPa and τₓᵧ = −10 MPa (tension positive). The element is rotated 45° clockwise. Find the normal stress σy′ on the faces of the rotated element that are perpendicular to the rotated y′-axis.

σx = −80 MPaσy = −70 MPaτxy = −10 MPaθ = -45°
σy′ = MPa
4.Medium
σx​ = 30σy​ = 80τxy​ = 10MPa

A point in a loaded part has σₓ = 30 MPa, σᵧ = 80 MPa and τₓᵧ = 10 MPa (tension positive). Find the maximum in-plane shear stress τmax.

σx = 30 MPaσy = 80 MPaτxy = 10 MPa
τmax = MPa
5.Hard
σx​ = 140σy​ = 10τxy​ = 10MPa

A point in a thin plate is in plane stress: σₓ = 140 MPa, σᵧ = 10 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 = 140 MPaσy = 10 MPaτxy = 10 MPaσY = 355 MPa
n =
6.Hard
σx​ = 30σy​ = 10τxy​ = −10MPa

A point in a thin plate is in plane stress: σₓ = 30 MPa, σᵧ = 10 MPa and τₓᵧ = −10 MPa (σz = 0). The material has a yield strength of 350 MPa. Find the factor of safety n against yielding by the Tresca (maximum shear stress) criterion.

σx = 30 MPaσy = 10 MPaτxy = −10 MPaσY = 350 MPa
n =

Answer key

  1. 1. σy′ = 111.8 MPa
  2. 2. σ1 = 107.1 MPa
  3. 3. σy′ = −85 MPa
  4. 4. τmax = 26.93 MPa
  5. 5. n = 2.603
  6. 6. n = 10.25

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