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​ = 30MPapure shear

A point on the surface of a shaft in torsion is in pure shear with τₓᵧ = 30 MPa (σₓ = σᵧ = 0). Find the normal stress σθ on a plane whose normal is rotated 45° clockwise from the x-axis.

τxy = 30 MPaθ = -45°
σθ = MPa
2.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
3.Medium
σx​ = −20σy​ = −10τxy​ = 20MPa

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

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

A point has σₓ = 0 MPa, σᵧ = −60 MPa and τₓᵧ = −10 MPa (tension positive). The element is rotated 30° counter-clockwise. Find the normal stress σy′ on the faces of the rotated element that are perpendicular to the rotated y′-axis.

σx = 0 MPaσy = −60 MPaτxy = −10 MPaθ = 30°
σy′ = MPa
5.Hard
σx​ = 160σy​ = 110τxy​ = 20MPa

A point in a thin plate is in plane stress: σₓ = 160 MPa, σᵧ = 110 MPa and τₓᵧ = 20 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 = 160 MPaσy = 110 MPaτxy = 20 MPaσY = 350 MPa
n =
6.Hard
σx​ = −70σy​ = −10τxy​ = 30MPa

A point in a loaded part has σₓ = −70 MPa, σᵧ = −10 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 = −70 MPaσy = −10 MPaτxy = 30 MPaθ = -20°
σθ = MPa

Answer key

  1. 1. σθ = −30 MPa
  2. 2. σavg = 55 MPa
  3. 3. τmax = 20.62 MPa
  4. 4. σy′ = −36.34 MPa
  5. 5. n = 2.096
  6. 6. σθ = −82.26 MPa

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