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 40° counter-clockwise from the x-axis.

τxy = 60 MPaθ = 40°
σθ = MPa
2.Easy
20 kN20 kNθ = 45°A = 800 mm²

A bar with cross-sectional area 800 mm² carries an axial tensile force of 20 kN. Find the magnitude of the shear stress |τθ| on a section whose normal makes an angle of 45° with the bar axis.

P = 20 kNA = 800 mm²θ = 45°
|τθ| = MPa
3.Medium
σx​ = 10σy​ = 110τxy​ = −50MPa

A point has σₓ = 10 MPa, σᵧ = 110 MPa and τₓᵧ = −50 MPa (tension positive). The element is rotated 40° counter-clockwise. Find the normal stress σy′ on the faces of the rotated element that are perpendicular to the rotated y′-axis.

σx = 10 MPaσy = 110 MPaτxy = −50 MPaθ = 40°
σy′ = MPa
4.Medium
σx​ = 10σy​ = −30τxy​ = 60MPa

A point in a loaded part has σₓ = 10 MPa, σᵧ = −30 MPa and τₓᵧ = 60 MPa (tension positive). Find the maximum in-plane shear stress τmax.

σx = 10 MPaσy = −30 MPaτxy = 60 MPa
τmax = MPa
5.Hard
σx​ = −80σy​ = 0τxy​ = −40MPa

A point in a loaded part has σₓ = −80 MPa, σᵧ = 0 MPa and τₓᵧ = −40 MPa (tension positive). Find the normal stress σθ on a plane whose normal is rotated 20° clockwise from the x-axis.

σx = −80 MPaσy = 0 MPaτxy = −40 MPaθ = -20°
σθ = MPa
6.Hard
σx​ = 30σy​ = 10τxy​ = −30MPa

A point in a loaded part has σₓ = 30 MPa, σᵧ = 10 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 = 30 MPaσy = 10 MPaτxy = −30 MPaθ = 35°
σθ = MPa

Answer key

  1. 1. σθ = 59.09 MPa
  2. 2. |τθ| = 12.5 MPa
  3. 3. σy′ = 117.9 MPa
  4. 4. τmax = 63.25 MPa
  5. 5. σθ = −44.93 MPa
  6. 6. σθ = −4.771 MPa

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