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​ = −10σy​ = 20τxy​ = 60MPa

A point in a loaded part has σₓ = −10 MPa, σᵧ = 20 MPa and τₓᵧ = 60 MPa (tension positive). Find the largest principal stress σ1.

σx = −10 MPaσy = 20 MPaτxy = 60 MPa
σ1 = MPa
2.Easy
σx​ = −40σy​ = 20τxy​MPa

A point has σₓ = −40 MPa and σᵧ = 20 MPa, plus some shear stress. On an element rotated by 20° the normal stress in the rotated x′ direction is −7.3 MPa. Find the normal stress σy′ in the rotated y′ direction.

σx = −40 MPaσy = 20 MPaσx′ = −7.3 MPaθ = 20°
σy′ = MPa
3.Medium
σx​ = 30σy​ = 0τxy​ = 60MPa

A point in a loaded part has σₓ = 30 MPa, σᵧ = 0 MPa and τₓᵧ = 60 MPa (tension positive). Find the maximum in-plane shear stress τmax.

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

A point has σₓ = 100 MPa, σᵧ = 10 MPa and τₓᵧ = −40 MPa (tension positive). Find the angle θs, between −90° and 90°, from the x-axis to the normal of the plane on which the shear stress has its maximum positive value +R (shear positive in the same sense as τₓᵧ).

σx = 100 MPaσy = 10 MPaτxy = −40 MPa
θs = °
5.Hard
σx​ = −50σy​ = 40τxy​ = 10MPa

A point in a loaded part has σₓ = −50 MPa, σᵧ = 40 MPa and τₓᵧ = 10 MPa (tension positive). Find the normal stress σθ on a plane whose normal is rotated 30° counter-clockwise from the x-axis.

σx = −50 MPaσy = 40 MPaτxy = 10 MPaθ = 30°
σθ = MPa
6.Hard
35°D = 1200 mm, t = 12 mm, p = 0.5 MPap

A thin-walled cylindrical tank of inner diameter 1200 mm and wall thickness 12 mm holds gas at a gauge pressure of 0.5 MPa. It is made from plate welded along a helical seam at 35° to the axis of the cylinder. Find the magnitude of the shear stress |τn| acting along the weld seam.

D = 1200 mmt = 12 mmp = 0.5 MPaα = 35°
|τn| = MPa

Answer key

  1. 1. σ1 = 66.85 MPa
  2. 2. σy′ = −12.7 MPa
  3. 3. τmax = 61.85 MPa
  4. 4. θs = −65.82 °
  5. 5. σθ = −18.84 MPa
  6. 6. |τn| = 5.873 MPa

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