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​ = 160σy​ = 70τxy​ = −20MPa

A point in a loaded part has σₓ = 160 MPa, σᵧ = 70 MPa and τₓᵧ = −20 MPa (tension positive). Find the largest principal stress σ1.

σx = 160 MPaσy = 70 MPaτxy = −20 MPa
σ1 = MPa
2.Easy
σx​ = 0σy​ = −60τxy​ = −60MPa

A point in a loaded part has σₓ = 0 MPa, σᵧ = −60 MPa and τₓᵧ = −60 MPa (tension positive). Find the largest principal stress σ1.

σx = 0 MPaσy = −60 MPaτxy = −60 MPa
σ1 = MPa
3.Medium
σx​ = 130σy​ = −60τxy​ = 50MPa

A point has σₓ = 130 MPa, σᵧ = −60 MPa and τₓᵧ = 50 MPa (tension positive). The element is rotated 30° counter-clockwise. Find the normal stress σy′ on the faces of the rotated element that are perpendicular to the rotated y′-axis.

σx = 130 MPaσy = −60 MPaτxy = 50 MPaθ = 30°
σy′ = MPa
4.Medium
σx​ = −60σy​ = 60τxy​ = −30MPa

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

σx = −60 MPaσy = 60 MPaτxy = −30 MPa
τmax = MPa
5.Hard
σx​τxy​top surface of the shaft

A solid shaft of diameter 60 mm carries a bending moment of 300 N·m together with a torque of 750 N·m. At the top surface, where the bending stress is tensile and largest, find the largest principal stress σ1.

d = 60 mmM = 300 N·mT = 750 N·m
σ1 = MPa
6.Hard
σx​ = 30σy​ = 20τxy​ = 30MPa

A point in a loaded part has σₓ = 30 MPa, σᵧ = 20 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 = 30 MPaσy = 20 MPaτxy = 30 MPaθ = 15°
τθ = MPa

Answer key

  1. 1. σ1 = 164.2 MPa
  2. 2. σ1 = 37.08 MPa
  3. 3. σy′ = −55.8 MPa
  4. 4. τmax = 67.08 MPa
  5. 5. σ1 = 26.12 MPa
  6. 6. τθ = 23.48 MPa

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