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
τxy​ = 80MPapure shear

A point on the surface of a shaft in torsion is in pure shear with τₓᵧ = 80 MPa (σₓ = σᵧ = 0). Find the normal stress σθ on a plane whose normal is rotated 30° counter-clockwise from the x-axis.

τxy = 80 MPaθ = 30°
σθ = MPa
2.Easy
σx​ = −60σy​ = 50τxy​ = −40MPa

A point in a loaded part has σₓ = −60 MPa, σᵧ = 50 MPa and τₓᵧ = −40 MPa (tension positive). Find the largest principal stress σ1.

σx = −60 MPaσy = 50 MPaτxy = −40 MPa
σ1 = MPa
3.Medium
σx​ = 160σy​ = −60τxy​ = −30MPa

A point in a thin plate is in plane stress: σₓ = 160 MPa, σᵧ = −60 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 = 160 MPaσy = −60 MPaτxy = −30 MPaσz = 0
τabs = MPa
4.Medium
σx​ = −50σy​ = −20τxy​ = 20MPa

A point in a loaded part has σₓ = −50 MPa, σᵧ = −20 MPa and τₓᵧ = 20 MPa (tension positive). Find the maximum in-plane shear stress τmax.

σx = −50 MPaσy = −20 MPaτxy = 20 MPa
τmax = MPa
5.Hard
σx​ = 10σy​ = 70τxy​ = −40MPa

A point in a loaded part has σₓ = 10 MPa, σᵧ = 70 MPa and τₓᵧ = −40 MPa (tension positive). Find the normal stress σθ on a plane whose normal is rotated 30° counter-clockwise from the x-axis.

σx = 10 MPaσy = 70 MPaτxy = −40 MPaθ = 30°
σθ = MPa
6.Hard
σx​τxy​surface of the shaft

A solid shaft of diameter 50 mm carries an axial tensile force of 190 kN together with a torque of 2000 N·m. At the surface of the shaft, find the largest principal stress σ1.

d = 50 mmF = 190 kNT = 2000 N·m
σ1 = MPa

Answer key

  1. 1. σθ = 69.28 MPa
  2. 2. σ1 = 63.01 MPa
  3. 3. τabs = 114 MPa
  4. 4. τmax = 25 MPa
  5. 5. σθ = −9.641 MPa
  6. 6. σ1 = 143.2 MPa

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