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​ = 0σy​ = 10τxy​ = −40MPa

A point in a loaded part has σₓ = 0 MPa, σᵧ = 10 MPa and τₓᵧ = −40 MPa (tension positive). Find the smallest principal stress σ2.

σx = 0 MPaσy = 10 MPaτxy = −40 MPa
σ2 = MPa
2.Easy
10 kN10 kNθ = 70°A = 250 mm²

A bar with cross-sectional area 250 mm² carries an axial tensile force of 10 kN. Find the magnitude of the shear stress |τθ| on a section whose normal makes an angle of 70° with the bar axis.

P = 10 kNA = 250 mm²θ = 70°
|τθ| = MPa
3.Medium
σx​ = 140σy​ = −60τxy​ = −20MPa

A point in a loaded part has σₓ = 140 MPa, σᵧ = −60 MPa and τₓᵧ = −20 MPa (tension positive). Find the angle θp from the x-axis to the direction of the largest principal stress, between −90° and 90°.

σx = 140 MPaσy = −60 MPaτxy = −20 MPa
θp = °
4.Medium
σx​ = 60σy​ = 90τxy​ = 30MPa

A point has σₓ = 60 MPa, σᵧ = 90 MPa and τₓᵧ = 30 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 = 60 MPaσy = 90 MPaτxy = 30 MPa
θs = °
5.Hard
σx​τxy​top surface of the shaft

A solid shaft of diameter 30 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 = 30 mmM = 300 N·mT = 750 N·m
σ1 = MPa
6.Hard
σx​ = 80σy​ = 120τxy​ = −40MPa

A point in a loaded part has σₓ = 80 MPa, σᵧ = 120 MPa and τₓᵧ = −40 MPa (tension positive). Find the normal stress σθ on a plane whose normal is rotated 20° counter-clockwise from the x-axis.

σx = 80 MPaσy = 120 MPaτxy = −40 MPaθ = 20°
σθ = MPa

Answer key

  1. 1. σ2 = −35.31 MPa
  2. 2. |τθ| = 12.86 MPa
  3. 3. θp = −5.655 °
  4. 4. θs = 13.28 °
  5. 5. σ1 = 209 MPa
  6. 6. σθ = 58.97 MPa

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