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​ = 20σy​ = −50τxy​ = 30MPa

A point in a loaded part has σₓ = 20 MPa, σᵧ = −50 MPa and τₓᵧ = 30 MPa (tension positive). Find the smallest principal stress σ2.

σx = 20 MPaσy = −50 MPaτxy = 30 MPa
σ2 = MPa
2.Easy
σx​ = 10σy​ = 30τxy​ = −20MPa

A point in a loaded part has σₓ = 10 MPa, σᵧ = 30 MPa and τₓᵧ = −20 MPa (tension positive). Find the largest principal stress σ1.

σx = 10 MPaσy = 30 MPaτxy = −20 MPa
σ1 = MPa
3.Medium
σx​ = 100σy​ = 50τxy​ = −50MPa

A point has σₓ = 100 MPa, σᵧ = 50 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 = 100 MPaσy = 50 MPaτxy = −50 MPa
θs = °
4.Medium
σx​ = 140σy​ = −70τxy​ = −60MPa

A point has σₓ = 140 MPa, σᵧ = −70 MPa and τₓᵧ = −60 MPa (tension positive). The element is rotated 60° counter-clockwise. Find the normal stress σy′ on the faces of the rotated element that are perpendicular to the rotated y′-axis.

σx = 140 MPaσy = −70 MPaτxy = −60 MPaθ = 60°
σy′ = MPa
5.Hard
σx​ = 90σy​ = −50τxy​ = 10MPa

A point in a thin plate is in plane stress: σₓ = 90 MPa, σᵧ = −50 MPa and τₓᵧ = 10 MPa (σz = 0). The material has a yield strength of 350 MPa. Find the factor of safety n against yielding by the Tresca (maximum shear stress) criterion.

σx = 90 MPaσy = −50 MPaτxy = 10 MPaσY = 350 MPa
n =
6.Hard
σx​ = 140σy​ = 60τxy​ = 30MPa

A point in a thin plate is in plane stress: σₓ = 140 MPa, σᵧ = 60 MPa and τₓᵧ = 30 MPa (σz = 0). Find the von Mises equivalent stress σvM.

σx = 140 MPaσy = 60 MPaτxy = 30 MPa
σvM = MPa

Answer key

  1. 1. σ2 = −61.1 MPa
  2. 2. σ1 = 42.36 MPa
  3. 3. θs = −76.72 °
  4. 4. σy′ = 139.5 MPa
  5. 5. n = 2.475
  6. 6. σvM = 132.3 MPa

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