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​ = −20σy​ = 70τxy​ = 60MPa

A point in a loaded part has σₓ = −20 MPa, σᵧ = 70 MPa and τₓᵧ = 60 MPa (tension positive). Find the largest principal stress σ1.

σx = −20 MPaσy = 70 MPaτxy = 60 MPa
σ1 = MPa
2.Easy
τxy​ = −110MPapure shear

A point on the surface of a shaft in torsion is in pure shear with τₓᵧ = −110 MPa (σₓ = σᵧ = 0). Find the normal stress σθ on a plane whose normal is rotated 40° counter-clockwise from the x-axis.

τxy = −110 MPaθ = 40°
σθ = MPa
3.Medium
σx​ = −20σy​ = 70MPaτxy​ = ?

A point has σₓ = −20 MPa and σᵧ = 70 MPa. How large can the shear stress τxy (magnitude) become before the largest principal stress reaches 120 MPa?

σx = −20 MPaσy = 70 MPaσ1 = 120 MPa
τxy = MPa
4.Medium
σx​ = 60σy​ = −30τxy​ = −20MPa

A point in a loaded part has σₓ = 60 MPa, σᵧ = −30 MPa and τₓᵧ = −20 MPa (tension positive). Find the maximum in-plane shear stress τmax.

σx = 60 MPaσy = −30 MPaτxy = −20 MPa
τmax = MPa
5.Hard
55°D = 1000 mm, t = 10 mm, p = 2 MPap

A thin-walled cylindrical tank of inner diameter 1000 mm and wall thickness 10 mm holds gas at a gauge pressure of 2 MPa. It is made from plate welded along a helical seam at 55° to the axis of the cylinder. Find the normal stress σn acting across the weld seam.

D = 1000 mmt = 10 mmp = 2 MPaα = 55°
σn = MPa
6.Hard
σx​ = −70σy​ = 110τxy​ = −10MPa

A point in a loaded part has σₓ = −70 MPa, σᵧ = 110 MPa and τₓᵧ = −10 MPa (tension positive). Find the normal stress σθ on a plane whose normal is rotated 25° counter-clockwise from the x-axis.

σx = −70 MPaσy = 110 MPaτxy = −10 MPaθ = 25°
σθ = MPa

Answer key

  1. 1. σ1 = 100 MPa
  2. 2. σθ = −108.3 MPa
  3. 3. τxy = 83.67 MPa
  4. 4. τmax = 49.24 MPa
  5. 5. σn = 66.45 MPa
  6. 6. σθ = −45.51 MPa

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