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
τxy​ = −60MPapure shear

A point on the surface of a shaft in torsion is in pure shear with τₓᵧ = −60 MPa (σₓ = σᵧ = 0). Find the normal stress σθ on a plane whose normal is rotated 20° counter-clockwise from the x-axis.

τxy = −60 MPaθ = 20°
σθ = MPa
2.Easy
σx​ = 120σy​ = 30τxy​MPa

A point has σₓ = 120 MPa and σᵧ = 30 MPa, plus some shear stress. On an element rotated by 35° the normal stress in the rotated x′ direction is 99.8 MPa. Find the normal stress σy′ in the rotated y′ direction.

σx = 120 MPaσy = 30 MPaσx′ = 99.8 MPaθ = 35°
σy′ = MPa
3.Medium
σx​ = 10σy​ = −10τxy​ = −50MPa

A point in a loaded part has σₓ = 10 MPa, σᵧ = −10 MPa and τₓᵧ = −50 MPa (tension positive). Find the maximum in-plane shear stress τmax.

σx = 10 MPaσy = −10 MPaτxy = −50 MPa
τmax = MPa
4.Medium
σx​ = 80σy​ = −20τxy​ = 10MPa

A point in a loaded part has σₓ = 80 MPa, σᵧ = −20 MPa and τₓᵧ = 10 MPa (tension positive). Find the maximum in-plane shear stress τmax.

σx = 80 MPaσy = −20 MPaτxy = 10 MPa
τmax = MPa
5.Hard
σx​ = 150σy​ = −60τxy​ = 50MPa

A point in a thin plate is in plane stress: σₓ = 150 MPa, σᵧ = −60 MPa and τₓᵧ = 50 MPa (σz = 0). The material has a yield strength of 355 MPa. Find the factor of safety n against yielding by the Tresca (maximum shear stress) criterion.

σx = 150 MPaσy = −60 MPaτxy = 50 MPaσY = 355 MPa
n =
6.Hard
σx​ = 40σy​ = −20τxy​ = 30MPa

A point in a loaded part has σₓ = 40 MPa, σᵧ = −20 MPa and τₓᵧ = 30 MPa (tension positive). Find the normal stress σθ on a plane whose normal is rotated 20° clockwise from the x-axis.

σx = 40 MPaσy = −20 MPaτxy = 30 MPaθ = -20°
σθ = MPa

Answer key

  1. 1. σθ = −38.57 MPa
  2. 2. σy′ = 50.2 MPa
  3. 3. τmax = 50.99 MPa
  4. 4. τmax = 50.99 MPa
  5. 5. n = 1.526
  6. 6. σθ = 13.7 MPa

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