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​ = −70σy​ = −20τxy​ = −40MPa

A point in a loaded part has σₓ = −70 MPa, σᵧ = −20 MPa and τₓᵧ = −40 MPa (tension positive). Find the smallest principal stress σ2.

σx = −70 MPaσy = −20 MPaτxy = −40 MPa
σ2 = MPa
2.Easy
σx​ = −60σy​ = 10τxy​ = 10MPa

A point in a loaded part has σₓ = −60 MPa, σᵧ = 10 MPa and τₓᵧ = 10 MPa (tension positive). Find the largest principal stress σ1.

σx = −60 MPaσy = 10 MPaτxy = 10 MPa
σ1 = MPa
3.Medium
σx​ = −50σy​ = 120τxy​ = 10MPa

A point has σₓ = −50 MPa, σᵧ = 120 MPa and τₓᵧ = 10 MPa (tension positive). Find the angle θs, between −90° and 90°, from the x-axis to the normal of the plane on which the shear stress has its maximum positive value +R (shear positive in the same sense as τₓᵧ).

σx = −50 MPaσy = 120 MPaτxy = 10 MPa
θs = °
4.Medium
σx​ = 120σy​ = −20τxy​ = −20MPa

A point in a loaded part has σₓ = 120 MPa, σᵧ = −20 MPa and τₓᵧ = −20 MPa (tension positive). Find the maximum in-plane shear stress τmax.

σx = 120 MPaσy = −20 MPaτxy = −20 MPa
τmax = MPa
5.Hard
σx​ = 160σy​ = 20τxy​ = 20MPa

A point in a loaded part has σₓ = 160 MPa, σᵧ = 20 MPa and τₓᵧ = 20 MPa (tension positive). Find the normal stress σθ on a plane whose normal is rotated 20° clockwise from the x-axis.

σx = 160 MPaσy = 20 MPaτxy = 20 MPaθ = -20°
σθ = MPa
6.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 500 N·m. At the top surface, where the bending stress is tensile and largest, find the maximum in-plane shear stress τmax.

d = 50 mmM = 200 N·mT = 500 N·m
τmax = MPa

Answer key

  1. 1. σ2 = −92.17 MPa
  2. 2. σ1 = 11.4 MPa
  3. 3. θs = 41.65 °
  4. 4. τmax = 72.8 MPa
  5. 5. σθ = 130.8 MPa
  6. 6. τmax = 21.94 MPa

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