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​ = 160σy​ = 50MPaprincipal element

At a point in a loaded part the in-plane principal stresses are σ₁ = 160 MPa and σ₂ = 50 MPa (tension positive). Find the normal stress σavg that acts on the planes of maximum in-plane shear stress.

σ1 = 160 MPaσ2 = 50 MPa
σavg = MPa
2.Easy
σy​ = 100τxy​MPa

A point has σₓ = 0 MPa and σᵧ = 100 MPa, plus some shear stress. On an element rotated by 30° the normal stress in the rotated x′ direction is −9.6 MPa. Find the normal stress σy′ in the rotated y′ direction.

σx = 0 MPaσy = 100 MPaσx′ = −9.6 MPaθ = 30°
σy′ = MPa
3.Medium
σx​ = −20σy​ = 0τxy​ = −60MPa

A point in a loaded part has σₓ = −20 MPa, σᵧ = 0 MPa and τₓᵧ = −60 MPa (tension positive). Find the maximum in-plane shear stress τmax.

σx = −20 MPaσy = 0 MPaτxy = −60 MPa
τmax = MPa
4.Medium
σx​ = 10σy​ = −80τxy​ = −60MPa

A point has σₓ = 10 MPa, σᵧ = −80 MPa and τₓᵧ = −60 MPa (tension positive). The element is rotated 20° clockwise. Find the normal stress σy′ on the faces of the rotated element that are perpendicular to the rotated y′-axis.

σx = 10 MPaσy = −80 MPaτxy = −60 MPaθ = -20°
σy′ = MPa
5.Hard
σx​ = 100σy​ = 100τxy​ = −30MPa

A point in a loaded part has σₓ = 100 MPa, σᵧ = 100 MPa and τₓᵧ = −30 MPa (tension positive). Find the normal stress σθ on a plane whose normal is rotated 35° counter-clockwise from the x-axis.

σx = 100 MPaσy = 100 MPaτxy = −30 MPaθ = 35°
σθ = MPa
6.Hard
σx​ = 60σy​ = 100τxy​ = 30MPa

A point in a loaded part has σₓ = 60 MPa, σᵧ = 100 MPa and τₓᵧ = 30 MPa (tension positive). Find the normal stress σθ on a plane whose normal is rotated 15° counter-clockwise from the x-axis.

σx = 60 MPaσy = 100 MPaτxy = 30 MPaθ = 15°
σθ = MPa

Answer key

  1. 1. σavg = 105 MPa
  2. 2. σy′ = 109.6 MPa
  3. 3. τmax = 60.83 MPa
  4. 4. σy′ = −108 MPa
  5. 5. σθ = 71.81 MPa
  6. 6. σθ = 77.68 MPa

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