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​ = −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 35° counter-clockwise from the x-axis.

τxy = −110 MPaθ = 35°
σθ = MPa
2.Easy
σx​ = −30σy​ = 60τxy​MPa

A point has σₓ = −30 MPa and σᵧ = 60 MPa, plus some shear stress. On an element rotated by 50° the normal stress in the rotated x′ direction is 81.9 MPa. Find the normal stress σy′ in the rotated y′ direction.

σx = −30 MPaσy = 60 MPaσx′ = 81.9 MPaθ = 50°
σy′ = MPa
3.Medium
σx​ = 110σy​ = 30τxy​ = −60MPa

A point in a thin plate is in plane stress: σₓ = 110 MPa, σᵧ = 30 MPa and τₓᵧ = −60 MPa, with σz = 0. Find the absolute maximum shear stress τabs at the point (considering all planes in three dimensions).

σx = 110 MPaσy = 30 MPaτxy = −60 MPaσz = 0
τabs = MPa
4.Medium
σx​ = 40σy​ = 100τxy​ = −10MPa

A point in a loaded part has σₓ = 40 MPa, σᵧ = 100 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 = 40 MPaσy = 100 MPaτxy = −10 MPa
θp = °
5.Hard
30°D = 600 mm, t = 5 mm, p = 2.5 MPap

A thin-walled cylindrical tank of inner diameter 600 mm and wall thickness 5 mm holds gas at a gauge pressure of 2.5 MPa. It is made from plate welded along a helical seam at 30° to the axis of the cylinder. Find the normal stress σn acting across the weld seam.

D = 600 mmt = 5 mmp = 2.5 MPaα = 30°
σn = MPa
6.Hard
σx​ = −30σy​ = −50τxy​ = 10MPa

A point in a loaded part has σₓ = −30 MPa, σᵧ = −50 MPa and τₓᵧ = 10 MPa (tension positive). Find the normal stress σθ on a plane whose normal is rotated 30° clockwise from the x-axis.

σx = −30 MPaσy = −50 MPaτxy = 10 MPaθ = -30°
σθ = MPa

Answer key

  1. 1. σθ = −103.4 MPa
  2. 2. σy′ = −51.9 MPa
  3. 3. τabs = 72.11 MPa
  4. 4. θp = −80.78 °
  5. 5. σn = 131.3 MPa
  6. 6. σθ = −43.66 MPa

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