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​ = −60σy​ = 40τxy​ = −10MPa

A point in a loaded part has σₓ = −60 MPa, σᵧ = 40 MPa and τₓᵧ = −10 MPa (tension positive). Find the smallest principal stress σ2.

σx = −60 MPaσy = 40 MPaτxy = −10 MPa
σ2 = MPa
2.Easy
σx​ = −50σy​ = 50τxy​MPa

A point has σₓ = −50 MPa and σᵧ = 50 MPa, plus some shear stress. On an element rotated by 50° the normal stress in the rotated x′ direction is 38.2 MPa. Find the normal stress σy′ in the rotated y′ direction.

σx = −50 MPaσy = 50 MPaσx′ = 38.2 MPaθ = 50°
σy′ = MPa
3.Medium
σx​ = 160σy​ = −30τxy​ = 50MPa

A point in a loaded part has σₓ = 160 MPa, σᵧ = −30 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 = 160 MPaσy = −30 MPaτxy = 50 MPa
θp = °
4.Medium
σx​ = 140σy​ = 60τxy​ = −50MPa

A point has σₓ = 140 MPa, σᵧ = 60 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 = 140 MPaσy = 60 MPaτxy = −50 MPa
θs = °
5.Hard
σx​ = −60σy​ = −50τxy​ = −50MPa

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

σx = −60 MPaσy = −50 MPaτxy = −50 MPaθ = -45°
τθ = MPa
6.Hard
σx​ = 130σy​ = −50τxy​ = 30MPa

A point in a thin plate is in plane stress: σₓ = 130 MPa, σᵧ = −50 MPa and τₓᵧ = 30 MPa (σz = 0). Find the von Mises equivalent stress σvM.

σx = 130 MPaσy = −50 MPaτxy = 30 MPa
σvM = MPa

Answer key

  1. 1. σ2 = −60.99 MPa
  2. 2. σy′ = −38.2 MPa
  3. 3. θp = 13.88 °
  4. 4. θs = −70.67 °
  5. 5. τθ = −5 MPa
  6. 6. σvM = 169.1 MPa

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