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​ = 50σy​ = −30τxy​ = 30MPa

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

σx = 50 MPaσy = −30 MPaτxy = 30 MPa
σ2 = MPa
2.Easy
140 kN140 kNθ = 70°A = 1000 mm²

A bar with cross-sectional area 1000 mm² carries an axial compressive force of 140 kN. Find the normal stress σθ on a section whose normal makes an angle of 70° with the bar axis (tension positive).

P = −140 kNA = 1000 mm²θ = 70°
σθ = MPa
3.Medium
σx​ = −40σy​ = −60τxy​ = 30MPa

A point in a loaded part has σₓ = −40 MPa, σᵧ = −60 MPa and τₓᵧ = 30 MPa (tension positive). Find the maximum in-plane shear stress τmax.

σx = −40 MPaσy = −60 MPaτxy = 30 MPa
τmax = MPa
4.Medium
σx​ = −50σy​ = 10τxy​ = 60MPa

A point has σₓ = −50 MPa, σᵧ = 10 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 = −50 MPaσy = 10 MPaτxy = 60 MPa
θs = °
5.Hard
σx​ = 120σy​ = 90τxy​ = 30MPa

A point in a thin plate is in plane stress: σₓ = 120 MPa, σᵧ = 90 MPa and τₓᵧ = 30 MPa (σz = 0). The material has a yield strength of 460 MPa. Find the factor of safety n against yielding by the Tresca (maximum shear stress) criterion.

σx = 120 MPaσy = 90 MPaτxy = 30 MPaσY = 460 MPa
n =
6.Hard
σx​ = −80σy​ = −60τxy​ = 30MPa

A point in a loaded part has σₓ = −80 MPa, σᵧ = −60 MPa and τₓᵧ = 30 MPa (tension positive). Find the normal stress σθ on a plane whose normal is rotated 15° counter-clockwise from the x-axis.

σx = −80 MPaσy = −60 MPaτxy = 30 MPaθ = 15°
σθ = MPa

Answer key

  1. 1. σ2 = −40 MPa
  2. 2. σθ = −16.38 MPa
  3. 3. τmax = 31.62 MPa
  4. 4. θs = 13.28 °
  5. 5. n = 3.32
  6. 6. σθ = −63.66 MPa

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