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​ = 0σy​ = 70τxy​ = −60MPa

A point in a loaded part has σₓ = 0 MPa, σᵧ = 70 MPa and τₓᵧ = −60 MPa (tension positive). Find the smallest principal stress σ2.

σx = 0 MPaσy = 70 MPaτxy = −60 MPa
σ2 = MPa
2.Easy
σx​ = −40σy​ = 0τxy​ = 30MPa

A point in a loaded part has σₓ = −40 MPa, σᵧ = 0 MPa and τₓᵧ = 30 MPa (tension positive). Find the smallest principal stress σ2.

σx = −40 MPaσy = 0 MPaτxy = 30 MPa
σ2 = MPa
3.Medium
σx​ = 110σy​ = −80τxy​ = 20MPa

A point in a loaded part has σₓ = 110 MPa, σᵧ = −80 MPa and τₓᵧ = 20 MPa (tension positive). Find the angle θp from the x-axis to the direction of the largest principal stress, between −90° and 90°.

σx = 110 MPaσy = −80 MPaτxy = 20 MPa
θp = °
4.Medium
σy​ = −30τxy​ = −20MPa

A point has σₓ = 0 MPa, σᵧ = −30 MPa and τₓᵧ = −20 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 = 0 MPaσy = −30 MPaτxy = −20 MPa
θs = °
5.Hard
σx​ = 120σy​ = 110τxy​ = 10MPa

A point in a loaded part has σₓ = 120 MPa, σᵧ = 110 MPa and τₓᵧ = 10 MPa (tension positive). Find the normal stress σθ on a plane whose normal is rotated 45° clockwise from the x-axis.

σx = 120 MPaσy = 110 MPaτxy = 10 MPaθ = -45°
σθ = MPa
6.Hard
60°D = 800 mm, t = 12 mm, p = 2 MPap

A thin-walled cylindrical tank of inner diameter 800 mm and wall thickness 12 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 magnitude of the shear stress |τn| acting along the weld seam.

D = 800 mmt = 12 mmp = 2 MPaα = 60°
|τn| = MPa

Answer key

  1. 1. σ2 = −34.46 MPa
  2. 2. σ2 = −56.06 MPa
  3. 3. θp = 5.944 °
  4. 4. θs = −71.57 °
  5. 5. σθ = 105 MPa
  6. 6. |τn| = 14.43 MPa

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