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​ = −40σy​ = 100τxy​ = 20MPa

A point in a loaded part has σₓ = −40 MPa, σᵧ = 100 MPa and τₓᵧ = 20 MPa (tension positive). Find the smallest principal stress σ2.

σx = −40 MPaσy = 100 MPaτxy = 20 MPa
σ2 = MPa
2.Easy
σx​ = 70σy​ = 10τxy​MPa

A point has σₓ = 70 MPa and σᵧ = 10 MPa, plus some shear stress. On an element rotated by 20° the normal stress in the rotated x′ direction is 50.1 MPa. Find the normal stress σy′ in the rotated y′ direction.

σx = 70 MPaσy = 10 MPaσx′ = 50.1 MPaθ = 20°
σy′ = MPa
3.Medium
σx​ = −40σy​ = −30MPaτxy​ = ?

A point has σₓ = −40 MPa and σᵧ = −30 MPa. How large can the shear stress τxy (magnitude) become before the largest principal stress reaches −15 MPa?

σx = −40 MPaσy = −30 MPaσ1 = −15 MPa
τxy = MPa
4.Medium
σx​ = 70σy​ = 40τxy​ = 40MPa

A point has σₓ = 70 MPa, σᵧ = 40 MPa and τₓᵧ = 40 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 = 70 MPaσy = 40 MPaτxy = 40 MPaθ = 25°
σy′ = MPa
5.Hard
σx​ = −50σy​ = 80τxy​ = 50MPa

A point in a thin plate is in plane stress: σₓ = −50 MPa, σᵧ = 80 MPa and τₓᵧ = 50 MPa (σz = 0). The material has a yield strength of 300 MPa. Find the factor of safety n against yielding by the Tresca (maximum shear stress) criterion.

σx = −50 MPaσy = 80 MPaτxy = 50 MPaσY = 300 MPa
n =
6.Hard
σx​ = −50σy​ = −20τxy​ = 60MPa

A point in a loaded part has σₓ = −50 MPa, σᵧ = −20 MPa and τₓᵧ = 60 MPa (tension positive). Find the shear stress τθ on a plane whose normal is rotated 45° counter-clockwise from the x-axis (positive in the same sense as τₓᵧ).

σx = −50 MPaσy = −20 MPaτxy = 60 MPaθ = 45°
τθ = MPa

Answer key

  1. 1. σ2 = −42.8 MPa
  2. 2. σy′ = 29.9 MPa
  3. 3. τxy = 19.36 MPa
  4. 4. σy′ = 14.72 MPa
  5. 5. n = 1.829
  6. 6. τθ = 15 MPa

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