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
50 kN50 kNθ = 30°A = 1000 mm²

A bar with cross-sectional area 1000 mm² carries an axial tensile force of 50 kN. Find the magnitude of the shear stress |τθ| on a section whose normal makes an angle of 30° with the bar axis.

P = 50 kNA = 1000 mm²θ = 30°
|τθ| = MPa
2.Easy
σy​ = 120τxy​MPa

A point has σₓ = 0 MPa and σᵧ = 120 MPa, plus some shear stress. On an element rotated by 60° the normal stress in the rotated x′ direction is 55.4 MPa. Find the normal stress σy′ in the rotated y′ direction.

σx = 0 MPaσy = 120 MPaσx′ = 55.4 MPaθ = 60°
σy′ = MPa
3.Medium
σx​ = −20σy​ = 10MPaτxy​ = ?

A point has σₓ = −20 MPa and σᵧ = 10 MPa. How large can the shear stress τxy (magnitude) become before the largest principal stress reaches 45 MPa?

σx = −20 MPaσy = 10 MPaσ1 = 45 MPa
τxy = MPa
4.Medium
σx​ = −40σy​ = −60τxy​ = 30MPa

A point in a loaded part has σₓ = −40 MPa, σᵧ = −60 MPa and τₓᵧ = 30 MPa (tension positive). Find the maximum in-plane shear stress τmax.

σx = −40 MPaσy = −60 MPaτxy = 30 MPa
τmax = MPa
5.Hard
σx​ = 110σy​ = −20τxy​ = −50MPa

A point in a loaded part has σₓ = 110 MPa, σᵧ = −20 MPa and τₓᵧ = −50 MPa (tension positive). Find the shear stress τθ on a plane whose normal is rotated 50° counter-clockwise from the x-axis (positive in the same sense as τₓᵧ).

σx = 110 MPaσy = −20 MPaτxy = −50 MPaθ = 50°
τθ = MPa
6.Hard
σx​ = 90σy​ = 70τxy​ = −30MPa

A point in a loaded part has σₓ = 90 MPa, σᵧ = 70 MPa and τₓᵧ = −30 MPa (tension positive). Find the shear stress τθ on a plane whose normal is rotated 15° counter-clockwise from the x-axis (positive in the same sense as τₓᵧ).

σx = 90 MPaσy = 70 MPaτxy = −30 MPaθ = 15°
τθ = MPa

Answer key

  1. 1. |τθ| = 21.65 MPa
  2. 2. σy′ = 64.6 MPa
  3. 3. τxy = 47.7 MPa
  4. 4. τmax = 31.62 MPa
  5. 5. τθ = −55.33 MPa
  6. 6. τθ = −30.98 MPa

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