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
30 kN30 kNθ = 25°A = 400 mm²

A bar with cross-sectional area 400 mm² carries an axial compressive force of 30 kN. Find the magnitude of the shear stress |τθ| on a section whose normal makes an angle of 25° with the bar axis.

P = −30 kNA = 400 mm²θ = 25°
|τθ| = MPa
2.Easy
τxy​ = 90MPapure shear

A point on the surface of a shaft in torsion is in pure shear with τₓᵧ = 90 MPa (σₓ = σᵧ = 0). Find the normal stress σθ on a plane whose normal is rotated 40° counter-clockwise from the x-axis.

τxy = 90 MPaθ = 40°
σθ = MPa
3.Medium
σx​ = −20σy​ = −20τxy​ = −10MPa

A point in a thin plate is in plane stress: σₓ = −20 MPa, σᵧ = −20 MPa and τₓᵧ = −10 MPa, with σz = 0. Find the absolute maximum shear stress τabs at the point (considering all planes in three dimensions).

σx = −20 MPaσy = −20 MPaτxy = −10 MPaσz = 0
τabs = MPa
4.Medium
σx​ = −10σy​ = 20τxy​ = −20MPa

A point in a loaded part has σₓ = −10 MPa, σᵧ = 20 MPa and τₓᵧ = −20 MPa (tension positive). Find the angle θp from the x-axis to the direction of the largest principal stress, between −90° and 90°.

σx = −10 MPaσy = 20 MPaτxy = −20 MPa
θp = °
5.Hard
σx​ = −50σy​ = 30τxy​ = −40MPa

A point in a thin plate is in plane stress: σₓ = −50 MPa, σᵧ = 30 MPa and τₓᵧ = −40 MPa (σz = 0). The material has a yield strength of 250 MPa. Find the factor of safety n against yielding by the von Mises (distortion energy) criterion.

σx = −50 MPaσy = 30 MPaτxy = −40 MPaσY = 250 MPa
n =
6.Hard
σx​ = −30σy​ = −60τxy​ = −20MPa

A point in a thin plate is in plane stress: σₓ = −30 MPa, σᵧ = −60 MPa and τₓᵧ = −20 MPa (σz = 0). The material has a yield strength of 460 MPa. Find the factor of safety n against yielding by the von Mises (distortion energy) criterion.

σx = −30 MPaσy = −60 MPaτxy = −20 MPaσY = 460 MPa
n =

Answer key

  1. 1. |τθ| = 28.73 MPa
  2. 2. σθ = 88.63 MPa
  3. 3. τabs = 15 MPa
  4. 4. θp = −63.43 °
  5. 5. n = 2.538
  6. 6. n = 7.366

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