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​ = −60σy​ = 60τxy​ = −60MPa

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

σx = −60 MPaσy = 60 MPaτxy = −60 MPa
σ1 = MPa
2.Easy
σx​ = 110σy​ = −10MPaprincipal element

At a point in a loaded part the in-plane principal stresses are σ₁ = 110 MPa and σ₂ = −10 MPa (tension positive). Find the maximum in-plane shear stress τmax.

σ1 = 110 MPaσ2 = −10 MPa
τmax = MPa
3.Medium
σx​ = 130σy​ = 80MPaτxy​ = ?

A point has σₓ = 130 MPa and σᵧ = 80 MPa. How large can the shear stress τxy (magnitude) become before the largest principal stress reaches 145 MPa?

σx = 130 MPaσy = 80 MPaσ1 = 145 MPa
τxy = MPa
4.Medium
σx​ = 150σy​ = 110τxy​ = −20MPa

A point has σₓ = 150 MPa, σᵧ = 110 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 = 150 MPaσy = 110 MPaτxy = −20 MPa
θs = °
5.Hard
50°D = 1200 mm, t = 6 mm, p = 0.5 MPap

A thin-walled cylindrical tank of inner diameter 1200 mm and wall thickness 6 mm holds gas at a gauge pressure of 0.5 MPa. It is made from plate welded along a helical seam at 50° to the axis of the cylinder. Find the magnitude of the shear stress |τn| acting along the weld seam.

D = 1200 mmt = 6 mmp = 0.5 MPaα = 50°
|τn| = MPa
6.Hard
25°D = 1200 mm, t = 8 mm, p = 1.2 MPap

A thin-walled cylindrical tank of inner diameter 1200 mm and wall thickness 8 mm holds gas at a gauge pressure of 1.2 MPa. It is made from plate welded along a helical seam at 25° to the axis of the cylinder. Find the normal stress σn acting across the weld seam.

D = 1200 mmt = 8 mmp = 1.2 MPaα = 25°
σn = MPa

Answer key

  1. 1. σ1 = 84.85 MPa
  2. 2. τmax = 60 MPa
  3. 3. τxy = 31.22 MPa
  4. 4. θs = −67.5 °
  5. 5. |τn| = 12.31 MPa
  6. 6. σn = 81.96 MPa

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