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
30 kN30 kNθ = 45°A = 250 mm²

A bar with cross-sectional area 250 mm² carries an axial compressive force of 30 kN. Find the normal stress σθ on a section whose normal makes an angle of 45° with the bar axis (tension positive).

P = −30 kNA = 250 mm²θ = 45°
σθ = MPa
2.Easy
140 kN140 kNθ = 30°A = 400 mm²

A bar with cross-sectional area 400 mm² carries an axial compressive force of 140 kN. Find the normal stress σθ on a section whose normal makes an angle of 30° with the bar axis (tension positive).

P = −140 kNA = 400 mm²θ = 30°
σθ = MPa
3.Medium
σx​ = 110σy​ = −50τxy​ = 30MPa

A point in a loaded part has σₓ = 110 MPa, σᵧ = −50 MPa and τₓᵧ = 30 MPa (tension positive). Find the angle θp from the x-axis to the direction of the largest principal stress, between −90° and 90°.

σx = 110 MPaσy = −50 MPaτxy = 30 MPa
θp = °
4.Medium
σx​ = 40σy​ = −20τxy​ = −50MPa

A point has σₓ = 40 MPa, σᵧ = −20 MPa and τₓᵧ = −50 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 = 40 MPaσy = −20 MPaτxy = −50 MPa
θs = °
5.Hard
σx​ = 20σy​ = −40τxy​ = −40MPa

A point in a thin plate is in plane stress: σₓ = 20 MPa, σᵧ = −40 MPa and τₓᵧ = −40 MPa (σz = 0). Find the von Mises equivalent stress σvM.

σx = 20 MPaσy = −40 MPaτxy = −40 MPa
σvM = MPa
6.Hard
σx​ = −80σy​ = −40τxy​ = −50MPa

A point in a loaded part has σₓ = −80 MPa, σᵧ = −40 MPa and τₓᵧ = −50 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 = −40 MPaτxy = −50 MPaθ = 20°
σθ = MPa

Answer key

  1. 1. σθ = −60 MPa
  2. 2. σθ = −262.5 MPa
  3. 3. θp = 10.28 °
  4. 4. θs = −74.52 °
  5. 5. σvM = 87.18 MPa
  6. 6. σθ = −107.5 MPa

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