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
90 kN90 kNθ = 60°A = 800 mm²

A bar with cross-sectional area 800 mm² carries an axial compressive force of 90 kN. Find the normal stress σθ on a section whose normal makes an angle of 60° with the bar axis (tension positive).

P = −90 kNA = 800 mm²θ = 60°
σθ = MPa
2.Easy
30 kN30 kNθ = 25°A = 250 mm²

A bar with cross-sectional area 250 mm² carries an axial tensile force of 30 kN. Find the normal stress σθ on a section whose normal makes an angle of 25° with the bar axis (tension positive).

P = 30 kNA = 250 mm²θ = 25°
σθ = MPa
3.Medium
σx​ = 130σy​ = −30τxy​ = 30MPa

A point in a loaded part has σₓ = 130 MPa, σᵧ = −30 MPa and τₓᵧ = 30 MPa (tension positive). Find the maximum in-plane shear stress τmax.

σx = 130 MPaσy = −30 MPaτxy = 30 MPa
τmax = MPa
4.Medium
σx​ = 50σy​ = 100τxy​ = −50MPa

A point in a loaded part has σₓ = 50 MPa, σᵧ = 100 MPa and τₓᵧ = −50 MPa (tension positive). Find the maximum in-plane shear stress τmax.

σx = 50 MPaσy = 100 MPaτxy = −50 MPa
τmax = MPa
5.Hard
σx​τxy​top surface of the shaft

A solid shaft of diameter 50 mm carries a bending moment of 200 N·m together with a torque of 1500 N·m. At the top surface, where the bending stress is tensile and largest, find the largest principal stress σ1.

d = 50 mmM = 200 N·mT = 1500 N·m
σ1 = MPa
6.Hard
σx​ = −60σy​ = 10τxy​ = 30MPa

A point in a thin plate is in plane stress: σₓ = −60 MPa, σᵧ = 10 MPa and τₓᵧ = 30 MPa (σz = 0). The material has a yield strength of 460 MPa. Find the factor of safety n against yielding by the von Mises (distortion energy) criterion.

σx = −60 MPaσy = 10 MPaτxy = 30 MPaσY = 460 MPa
n =

Answer key

  1. 1. σθ = −28.13 MPa
  2. 2. σθ = 98.57 MPa
  3. 3. τmax = 85.44 MPa
  4. 4. τmax = 55.9 MPa
  5. 5. σ1 = 69.81 MPa
  6. 6. n = 5.498

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