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
190 kN190 kNθ = 30°A = 800 mm²

A bar with cross-sectional area 800 mm² carries an axial tensile force of 190 kN. Find the magnitude of the shear stress |τθ| on a section whose normal makes an angle of 30° with the bar axis.

P = 190 kNA = 800 mm²θ = 30°
|τθ| = MPa
2.Easy
σx​ = 50σy​ = −60τxy​ = 40MPa

A point in a loaded part has σₓ = 50 MPa, σᵧ = −60 MPa and τₓᵧ = 40 MPa (tension positive). Find the largest principal stress σ1.

σx = 50 MPaσy = −60 MPaτxy = 40 MPa
σ1 = MPa
3.Medium
σx​ = −80σy​ = −60τxy​ = 30MPa

A point in a thin plate is in plane stress: σₓ = −80 MPa, σᵧ = −60 MPa and τₓᵧ = 30 MPa, with σz = 0. Find the absolute maximum shear stress τabs at the point (considering all planes in three dimensions).

σx = −80 MPaσy = −60 MPaτxy = 30 MPaσz = 0
τabs = MPa
4.Medium
σx​ = −20σy​ = 90MPaτxy​ = ?

A point has σₓ = −20 MPa and σᵧ = 90 MPa. How large can the shear stress τxy (magnitude) become before the largest principal stress reaches 150 MPa?

σx = −20 MPaσy = 90 MPaσ1 = 150 MPa
τxy = MPa
5.Hard
σx​ = 60σy​ = −10τxy​ = −30MPa

A point in a loaded part has σₓ = 60 MPa, σᵧ = −10 MPa and τₓᵧ = −30 MPa (tension positive). Find the shear stress τθ on a plane whose normal is rotated 60° counter-clockwise from the x-axis (positive in the same sense as τₓᵧ).

σx = 60 MPaσy = −10 MPaτxy = −30 MPaθ = 60°
τθ = MPa
6.Hard
σx​ = 80σy​ = −80τxy​ = 50MPa

A point in a thin plate is in plane stress: σₓ = 80 MPa, σᵧ = −80 MPa and τₓᵧ = 50 MPa (σz = 0). The material has a yield strength of 460 MPa. Find the factor of safety n against yielding by the Tresca (maximum shear stress) criterion.

σx = 80 MPaσy = −80 MPaτxy = 50 MPaσY = 460 MPa
n =

Answer key

  1. 1. |τθ| = 102.8 MPa
  2. 2. σ1 = 63.01 MPa
  3. 3. τabs = 50.81 MPa
  4. 4. τxy = 101 MPa
  5. 5. τθ = −15.31 MPa
  6. 6. n = 2.438

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