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​ = −120MPapure shear

A point on the surface of a shaft in torsion is in pure shear with τₓᵧ = −120 MPa (σₓ = σᵧ = 0). Find the normal stress σθ on a plane whose normal is rotated 20° counter-clockwise from the x-axis.

τxy = −120 MPaθ = 20°
σθ = MPa
2.Easy
σx​ = 150σy​ = −70τxy​ = 20MPa

A point in a loaded part has σₓ = 150 MPa, σᵧ = −70 MPa and τₓᵧ = 20 MPa (tension positive). Find the largest principal stress σ1.

σx = 150 MPaσy = −70 MPaτxy = 20 MPa
σ1 = MPa
3.Medium
σx​ = 120σy​ = 90τxy​ = 20MPa

A point in a loaded part has σₓ = 120 MPa, σᵧ = 90 MPa and τₓᵧ = 20 MPa (tension positive). Find the maximum in-plane shear stress τmax.

σx = 120 MPaσy = 90 MPaτxy = 20 MPa
τmax = MPa
4.Medium
σx​ = −40σy​ = 70MPaτxy​ = ?

A point has σₓ = −40 MPa and σᵧ = 70 MPa. How large can the shear stress τxy (magnitude) become before the largest principal stress reaches 120 MPa?

σx = −40 MPaσy = 70 MPaσ1 = 120 MPa
τxy = MPa
5.Hard
20°D = 500 mm, t = 12 mm, p = 1 MPap

A thin-walled cylindrical tank of inner diameter 500 mm and wall thickness 12 mm holds gas at a gauge pressure of 1 MPa. It is made from plate welded along a helical seam at 20° to the axis of the cylinder. Find the normal stress σn acting across the weld seam.

D = 500 mmt = 12 mmp = 1 MPaα = 20°
σn = MPa
6.Hard
60°D = 800 mm, t = 8 mm, p = 2 MPap

A thin-walled cylindrical tank of inner diameter 800 mm and wall thickness 8 mm holds gas at a gauge pressure of 2 MPa. It is made from plate welded along a helical seam at 60° to the axis of the cylinder. Find the normal stress σn acting across the weld seam.

D = 800 mmt = 8 mmp = 2 MPaα = 60°
σn = MPa

Answer key

  1. 1. σθ = −77.13 MPa
  2. 2. σ1 = 151.8 MPa
  3. 3. τmax = 25 MPa
  4. 4. τxy = 89.44 MPa
  5. 5. σn = 19.61 MPa
  6. 6. σn = 62.5 MPa

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