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​ = −20σy​ = −50τxy​ = −60MPa

A point in a loaded part has σₓ = −20 MPa, σᵧ = −50 MPa and τₓᵧ = −60 MPa (tension positive). Find the largest principal stress σ1.

σx = −20 MPaσy = −50 MPaτxy = −60 MPa
σ1 = MPa
2.Easy
σx​ = 170σy​ = 80MPaprincipal element

At a point in a loaded part the in-plane principal stresses are σ₁ = 170 MPa and σ₂ = 80 MPa (tension positive). Find the maximum in-plane shear stress τmax.

σ1 = 170 MPaσ2 = 80 MPa
τmax = MPa
3.Medium
σx​ = 20σy​ = −30MPaτxy​ = ?

A point has σₓ = 20 MPa and σᵧ = −30 MPa. How large can the shear stress τxy (magnitude) become before the largest principal stress reaches 70 MPa?

σx = 20 MPaσy = −30 MPaσ1 = 70 MPa
τxy = MPa
4.Medium
σx​ = −70σy​ = 80τxy​ = 50MPa

A point in a loaded part has σₓ = −70 MPa, σᵧ = 80 MPa and τₓᵧ = 50 MPa (tension positive). Find the maximum in-plane shear stress τmax.

σx = −70 MPaσy = 80 MPaτxy = 50 MPa
τmax = MPa
5.Hard
55°D = 800 mm, t = 12 mm, p = 0.8 MPap

A thin-walled cylindrical tank of inner diameter 800 mm and wall thickness 12 mm holds gas at a gauge pressure of 0.8 MPa. It is made from plate welded along a helical seam at 55° to the axis of the cylinder. Find the magnitude of the shear stress |τn| acting along the weld seam.

D = 800 mmt = 12 mmp = 0.8 MPaα = 55°
|τn| = MPa
6.Hard
σx​τxy​surface of the shaft

A solid shaft of diameter 25 mm carries an axial compressive force of 180 kN together with a torque of 250 N·m. At the surface of the shaft, find the maximum in-plane shear stress τmax.

d = 25 mmF = −180 kNT = 250 N·m
τmax = MPa

Answer key

  1. 1. σ1 = 26.85 MPa
  2. 2. τmax = 45 MPa
  3. 3. τxy = 70.71 MPa
  4. 4. τmax = 90.14 MPa
  5. 5. |τn| = 6.265 MPa
  6. 6. τmax = 200.6 MPa

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