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
80 kN80 kNθ = 50°A = 400 mm²

A bar with cross-sectional area 400 mm² carries an axial compressive force of 80 kN. Find the normal stress σθ on a section whose normal makes an angle of 50° with the bar axis (tension positive).

P = −80 kNA = 400 mm²θ = 50°
σθ = MPa
2.Easy
σx​ = 110σy​ = −60τxy​MPa

A point has σₓ = 110 MPa and σᵧ = −60 MPa, plus some shear stress. On an element rotated by 60° the normal stress in the rotated x′ direction is −34.8 MPa. Find the normal stress σy′ in the rotated y′ direction.

σx = 110 MPaσy = −60 MPaσx′ = −34.8 MPaθ = 60°
σy′ = MPa
3.Medium
σx​ = −80σy​ = −50τxy​ = 30MPa

A point in a thin plate is in plane stress: σₓ = −80 MPa, σᵧ = −50 MPa and τₓᵧ = 30 MPa, with σz = 0. Find the absolute maximum shear stress τabs at the point (considering all planes in three dimensions).

σx = −80 MPaσy = −50 MPaτxy = 30 MPaσz = 0
τabs = MPa
4.Medium
σx​ = −60σy​ = 50τxy​ = −60MPa

A point in a loaded part has σₓ = −60 MPa, σᵧ = 50 MPa and τₓᵧ = −60 MPa (tension positive). Find the maximum in-plane shear stress τmax.

σx = −60 MPaσy = 50 MPaτxy = −60 MPa
τmax = MPa
5.Hard
σx​ = 0σy​ = −80τxy​ = −50MPa

A point in a loaded part has σₓ = 0 MPa, σᵧ = −80 MPa and τₓᵧ = −50 MPa (tension positive). Find the normal stress σθ on a plane whose normal is rotated 30° clockwise from the x-axis.

σx = 0 MPaσy = −80 MPaτxy = −50 MPaθ = -30°
σθ = MPa
6.Hard
σx​ = 10σy​ = 20τxy​ = 20MPa

A point in a loaded part has σₓ = 10 MPa, σᵧ = 20 MPa and τₓᵧ = 20 MPa (tension positive). Find the normal stress σθ on a plane whose normal is rotated 45° counter-clockwise from the x-axis.

σx = 10 MPaσy = 20 MPaτxy = 20 MPaθ = 45°
σθ = MPa

Answer key

  1. 1. σθ = −82.64 MPa
  2. 2. σy′ = 84.8 MPa
  3. 3. τabs = 49.27 MPa
  4. 4. τmax = 81.39 MPa
  5. 5. σθ = 23.3 MPa
  6. 6. σθ = 35 MPa

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