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​ = 130σy​ = 20τxy​ = −20MPa

A point in a loaded part has σₓ = 130 MPa, σᵧ = 20 MPa and τₓᵧ = −20 MPa (tension positive). Find the smallest principal stress σ2.

σx = 130 MPaσy = 20 MPaτxy = −20 MPa
σ2 = MPa
2.Easy
σx​ = 50σy​ = 70τxy​MPa

A point has σₓ = 50 MPa and σᵧ = 70 MPa, plus some shear stress. On an element rotated by 15° the normal stress in the rotated x′ direction is 21.3 MPa. Find the normal stress σy′ in the rotated y′ direction.

σx = 50 MPaσy = 70 MPaσx′ = 21.3 MPaθ = 15°
σy′ = MPa
3.Medium
σx​ = 30σy​ = 120τxy​ = 30MPa

A point in a loaded part has σₓ = 30 MPa, σᵧ = 120 MPa and τₓᵧ = 30 MPa (tension positive). Find the maximum in-plane shear stress τmax.

σx = 30 MPaσy = 120 MPaτxy = 30 MPa
τmax = MPa
4.Medium
σx​ = 140σy​ = 90τxy​ = −10MPa

A point in a thin plate is in plane stress: σₓ = 140 MPa, σᵧ = 90 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 = 140 MPaσy = 90 MPaτxy = −10 MPaσz = 0
τabs = MPa
5.Hard
20°D = 400 mm, t = 6 mm, p = 1.2 MPap

A thin-walled cylindrical tank of inner diameter 400 mm and wall thickness 6 mm holds gas at a gauge pressure of 1.2 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 = 400 mmt = 6 mmp = 1.2 MPaα = 20°
σn = MPa
6.Hard
σx​ = −40σy​ = 90τxy​ = −30MPa

A point in a loaded part has σₓ = −40 MPa, σᵧ = 90 MPa and τₓᵧ = −30 MPa (tension positive). Find the normal stress σθ on a plane whose normal is rotated 25° counter-clockwise from the x-axis.

σx = −40 MPaσy = 90 MPaτxy = −30 MPaθ = 25°
σθ = MPa

Answer key

  1. 1. σ2 = 16.48 MPa
  2. 2. σy′ = 98.7 MPa
  3. 3. τmax = 54.08 MPa
  4. 4. τabs = 70.96 MPa
  5. 5. σn = 37.66 MPa
  6. 6. σθ = −39.76 MPa

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