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​ = 40τxy​ = 60MPa

A point in a loaded part has σₓ = −70 MPa, σᵧ = 40 MPa and τₓᵧ = 60 MPa (tension positive). Find the smallest principal stress σ2.

σx = −70 MPaσy = 40 MPaτxy = 60 MPa
σ2 = MPa
2.Easy
τxy​ = −60MPapure shear

A point on the surface of a shaft in torsion is in pure shear with τₓᵧ = −60 MPa (σₓ = σᵧ = 0). Find the normal stress σθ on a plane whose normal is rotated 35° counter-clockwise from the x-axis.

τxy = −60 MPaθ = 35°
σθ = MPa
3.Medium
σx​ = 130σy​ = −10τxy​ = 40MPa

A point has σₓ = 130 MPa, σᵧ = −10 MPa and τₓᵧ = 40 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 = 130 MPaσy = −10 MPaτxy = 40 MPa
θs = °
4.Medium
σx​ = −60σy​ = 80τxy​ = 50MPa

A point in a loaded part has σₓ = −60 MPa, σᵧ = 80 MPa and τₓᵧ = 50 MPa (tension positive). Find the maximum in-plane shear stress τmax.

σx = −60 MPaσy = 80 MPaτxy = 50 MPa
τmax = MPa
5.Hard
σx​τxy​top surface of the shaft

A solid shaft of diameter 30 mm carries a bending moment of 500 N·m together with a torque of 1000 N·m. At the top surface, where the bending stress is tensile and largest, find the largest principal stress σ1.

d = 30 mmM = 500 N·mT = 1000 N·m
σ1 = MPa
6.Hard
σx​ = 10σy​ = 20τxy​ = 20MPa

A point in a loaded part has σₓ = 10 MPa, σᵧ = 20 MPa and τₓᵧ = 20 MPa (tension positive). Find the normal stress σθ on a plane whose normal is rotated 15° counter-clockwise from the x-axis.

σx = 10 MPaσy = 20 MPaτxy = 20 MPaθ = 15°
σθ = MPa

Answer key

  1. 1. σ2 = −96.39 MPa
  2. 2. σθ = −56.38 MPa
  3. 3. θs = −30.13 °
  4. 4. τmax = 86.02 MPa
  5. 5. σ1 = 305.2 MPa
  6. 6. σθ = 20.67 MPa

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