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
σx​ = 0σy​ = 90τxy​ = −50MPa

A point in a loaded part has σₓ = 0 MPa, σᵧ = 90 MPa and τₓᵧ = −50 MPa (tension positive). Find the smallest principal stress σ2.

σx = 0 MPaσy = 90 MPaτxy = −50 MPa
σ2 = MPa
2.Easy
σx​ = 130σy​ = −50τxy​ = −10MPa

A point in a loaded part has σₓ = 130 MPa, σᵧ = −50 MPa and τₓᵧ = −10 MPa (tension positive). Find the largest principal stress σ1.

σx = 130 MPaσy = −50 MPaτxy = −10 MPa
σ1 = MPa
3.Medium
σx​ = −20σy​ = 110τxy​ = 10MPa

A point in a loaded part has σₓ = −20 MPa, σᵧ = 110 MPa and τₓᵧ = 10 MPa (tension positive). Find the angle θp from the x-axis to the direction of the largest principal stress, between −90° and 90°.

σx = −20 MPaσy = 110 MPaτxy = 10 MPa
θp = °
4.Medium
σx​ = 160σy​ = 90τxy​ = 50MPa

A point in a thin plate is in plane stress: σₓ = 160 MPa, σᵧ = 90 MPa and τₓᵧ = 50 MPa, with σz = 0. Find the absolute maximum shear stress τabs at the point (considering all planes in three dimensions).

σx = 160 MPaσy = 90 MPaτxy = 50 MPaσz = 0
τabs = MPa
5.Hard
σx​ = 110σy​ = 90τxy​ = 60MPa

A point in a loaded part has σₓ = 110 MPa, σᵧ = 90 MPa and τₓᵧ = 60 MPa (tension positive). Find the normal stress σθ on a plane whose normal is rotated 50° counter-clockwise from the x-axis.

σx = 110 MPaσy = 90 MPaτxy = 60 MPaθ = 50°
σθ = MPa
6.Hard
σx​ = 80σy​ = 0τxy​ = −30MPa

A point in a loaded part has σₓ = 80 MPa, σᵧ = 0 MPa and τₓᵧ = −30 MPa (tension positive). Find the shear stress τθ on a plane whose normal is rotated 35° counter-clockwise from the x-axis (positive in the same sense as τₓᵧ).

σx = 80 MPaσy = 0 MPaτxy = −30 MPaθ = 35°
τθ = MPa

Answer key

  1. 1. σ2 = −22.27 MPa
  2. 2. σ1 = 130.6 MPa
  3. 3. θp = 85.63 °
  4. 4. τabs = 93.02 MPa
  5. 5. σθ = 157.4 MPa
  6. 6. τθ = −47.85 MPa

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