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
τxy​ = −50MPapure shear

A point on the surface of a shaft in torsion is in pure shear with τₓᵧ = −50 MPa (σₓ = σᵧ = 0). Find the normal stress σθ on a plane whose normal is rotated 45° counter-clockwise from the x-axis.

τxy = −50 MPaθ = 45°
σθ = MPa
2.Easy
τxy​ = −110MPapure shear

A point on the surface of a shaft in torsion is in pure shear with τₓᵧ = −110 MPa (σₓ = σᵧ = 0). Find the normal stress σθ on a plane whose normal is rotated 25° counter-clockwise from the x-axis.

τxy = −110 MPaθ = 25°
σθ = MPa
3.Medium
σx​ = 110σy​ = −10τxy​ = 50MPa

A point in a loaded part has σₓ = 110 MPa, σᵧ = −10 MPa and τₓᵧ = 50 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 = −10 MPaτxy = 50 MPa
θp = °
4.Medium
σx​ = 110σy​ = −20τxy​ = 30MPa

A point has σₓ = 110 MPa, σᵧ = −20 MPa and τₓᵧ = 30 MPa (tension positive). The element is rotated 25° counter-clockwise. Find the normal stress σy′ on the faces of the rotated element that are perpendicular to the rotated y′-axis.

σx = 110 MPaσy = −20 MPaτxy = 30 MPaθ = 25°
σy′ = MPa
5.Hard
σx​ = 120σy​ = 100τxy​ = −20MPa

A point in a thin plate is in plane stress: σₓ = 120 MPa, σᵧ = 100 MPa and τₓᵧ = −20 MPa (σz = 0). The material has a yield strength of 350 MPa. Find the factor of safety n against yielding by the von Mises (distortion energy) criterion.

σx = 120 MPaσy = 100 MPaτxy = −20 MPaσY = 350 MPa
n =
6.Hard
σx​ = 50σy​ = 40τxy​ = −40MPa

A point in a thin plate is in plane stress: σₓ = 50 MPa, σᵧ = 40 MPa and τₓᵧ = −40 MPa (σz = 0). The material has a yield strength of 250 MPa. Find the factor of safety n against yielding by the von Mises (distortion energy) criterion.

σx = 50 MPaσy = 40 MPaτxy = −40 MPaσY = 250 MPa
n =

Answer key

  1. 1. σθ = −50 MPa
  2. 2. σθ = −84.26 MPa
  3. 3. θp = 19.9 °
  4. 4. σy′ = −19.76 MPa
  5. 5. n = 3.001
  6. 6. n = 3.01

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