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​ = 120σy​ = 100τxy​ = 40MPa

A point in a loaded part has σₓ = 120 MPa, σᵧ = 100 MPa and τₓᵧ = 40 MPa (tension positive). Find the largest principal stress σ1.

σx = 120 MPaσy = 100 MPaτxy = 40 MPa
σ1 = MPa
2.Easy
σx​ = 20σy​ = −10τxy​ = −20MPa

A point in a loaded part has σₓ = 20 MPa, σᵧ = −10 MPa and τₓᵧ = −20 MPa (tension positive). Find the smallest principal stress σ2.

σx = 20 MPaσy = −10 MPaτxy = −20 MPa
σ2 = MPa
3.Medium
σx​ = 0σy​ = 30τxy​ = 20MPa

A point in a loaded part has σₓ = 0 MPa, σᵧ = 30 MPa and τₓᵧ = 20 MPa (tension positive). Find the maximum in-plane shear stress τmax.

σx = 0 MPaσy = 30 MPaτxy = 20 MPa
τmax = MPa
4.Medium
σx​ = 70σy​ = −80τxy​ = −60MPa

A point has σₓ = 70 MPa, σᵧ = −80 MPa and τₓᵧ = −60 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 = 70 MPaσy = −80 MPaτxy = −60 MPa
θs = °
5.Hard
σx​ = 20σy​ = 10τxy​ = −20MPa

A point in a loaded part has σₓ = 20 MPa, σᵧ = 10 MPa and τₓᵧ = −20 MPa (tension positive). Find the normal stress σθ on a plane whose normal is rotated 45° counter-clockwise from the x-axis.

σx = 20 MPaσy = 10 MPaτxy = −20 MPaθ = 45°
σθ = MPa
6.Hard
σx​ = 20σy​ = 90τxy​ = 10MPa

A point in a thin plate is in plane stress: σₓ = 20 MPa, σᵧ = 90 MPa and τₓᵧ = 10 MPa (σz = 0). The material has a yield strength of 300 MPa. Find the factor of safety n against yielding by the von Mises (distortion energy) criterion.

σx = 20 MPaσy = 90 MPaτxy = 10 MPaσY = 300 MPa
n =

Answer key

  1. 1. σ1 = 151.2 MPa
  2. 2. σ2 = −20 MPa
  3. 3. τmax = 25 MPa
  4. 4. θs = −64.33 °
  5. 5. σθ = −5 MPa
  6. 6. n = 3.586

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